<Text-field style="Heading 1" layout="Heading 1">C\303\241lculo de V\303\241rias Vari\303\241veis Passo a Passo</Text-field>

<Text-field style="Heading 2" layout="Heading 2">O Comando MultiInt </Text-field>

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O comando MultiInt do pacote Student Multivariate Calculus ir\303\241 formular e calcular uma integral m\303\272ltipla iterada. Uma de suas op\303\247\303\265es de sa\303\255da \303\251 uma exibi\303\247\303\243o dos passos envolvidos na execu\303\247\303\243o do c\303\241lculo. Tabela 27 mostra o uso deste comando para calcular o volume da regi\303\243o

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Carregando Student:-MultivariateCalculus

with(Student:-MultivariateCalculus):

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IyIjPCIkMCI=

Tabela 27 Volume da regi\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= computada pelo uso do comando MultiInt

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A primeira linha da sa\303\255da \303\251 uma integral n\303\243o calculada; e a \303\272ltima, o valor da integral. A segunda linha mostra a integral mais externa ap\303\263s a integral interna ter sido calculada como a antiderivada com respeito a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Para esta antiderivada, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= foi mantido fixo. A antiderivada deve ser calculada nos limites da integral interna. A terceira linha mostra a integral externa completamente em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. A quarta linha \303\251 a antiderivada com respeito a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= que deve ser calculada nos limites da integral externa. O valor final \303\251 a \303\272ltima linha.

Esta ferramenta de integra\303\247\303\243o est\303\241 dispon\303\255vel como modelo de tarefa na Tabela 28.

Ferramentas\342\211\273Tarefas\342\211\273Pesquisar:

Calculus - Multivariate\342\211\273Integration\342\211\273Multiple Integration\342\211\273Cartesian 2-D

Integral Dupla Iterada em Coordenadas Cartesianas

Integrando:

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Regi\303\243o: 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IiIh

LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JC1GLDYlUSJiRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y5USdub3JtYWxGJ0YrRjs=

LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNictSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JkYrLUYwNiZGKy1JKG1hY3Rpb25HRiQ2JC1GMDYlLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lLHBsYWNlaG9sZGVyR1EldHJ1ZUYnRj0vJSthY3Rpb250eXBlR1E7bWFwbGVzb2Z0OnRhc2stcGxhY2Vob2xkZXJGJ0YrRj1GK0Y9RisvJStleGVjdXRhYmxlR1EmZmFsc2VGJ0Y9

IiIi

Integral Inerte: LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2Ji1GLDYlUSNkeUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GOVEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkMvJSlzdHJldGNoeUdGQy8lKnN5bW1ldHJpY0dGQy8lKGxhcmdlb3BHRkMvJS5tb3ZhYmxlbGltaXRzR0ZDLyUnYWNjZW50R0ZDLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUi1GLDYlUSNkeEYnRjVGOEY/RitGPw==

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

LUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQtRiM2JCwoIiIiRiwqJClJInhHRiciIiNGLCEiIiokKUkieUdGJ0YwRixGMS9GNDtGLSokKUYvI0YsRjBGLC9GLzsiIiFGLA==

Valor:

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

IyIjPCIkMCI=

C\303\241lculo Passo a Passo:

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

IyIjPCIkMCI=

LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNictSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JEYrLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0YrLyUrZXhlY3V0YWJsZUdRJmZhbHNlRidGMg==

Tabela 28 Accesso ao comando MultiInt atrav\303\251s de um modelo de tarefas

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 2" layout="Heading 2">Visualizando Regi\303\265es de Integra\303\247\303\243o</Text-field>

<Text-field style="Heading 3" layout="Heading 3">Coordenadas Cartesianas: 2D</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Integre 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 sobre a regi\303\243o plana limitada pelas curvas LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JJW1zdXBHRiQ2JS1GLDYlUSJ4RidGL0YyLUYjNiUtSSNtbkdGJDYkUSIyRidGOUYvRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjk= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GIzYlLUYsNiNRIUYnLUkmbXNxcnRHRiQ2Iy1GIzYkLUYsNiVRInhGJ0YvRjJGOUY5RlFGOQ==.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

O modelo de tarefas na Tabela 29 providencia uma solu\303\247\303\243o. A localiza\303\247\303\243o do modelo de tarefa \303\251 dada no topo da tabela.

Ferramentas\342\211\273Tarefas\342\211\273Pesquisar:

Calculus - Multivariate\342\211\273Integration\342\211\273Visualizing Regions of Integration\342\211\273Cartesian 2-D

Calcule 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 e construa o Gr\303\241fico de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=

Elemento de \303\201rea LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JlEjZEFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjZRJ25vcm1hbEYn

Select dA,dy dx,dx dyuse DocumentTools in IntOrder := module() local Y,q1,g,G,a,b,x,y,Psi,q2; export X; q1 := Int(Psi,[y=g(x)..G(x), x=a..b]); q2 := Int(Psi,[x=g(y)..G(y), y=a..b]); Y := GetProperty(ComboBox0,value); if Y = "dy dx" then X:=1; Do(%MathContainer92 = q1); elif Y = "dx dy" then X:=2; Do(%MathContainer92 = q2); elif Y = "Select dA" then X:=3; Do(%MathContainer92 = NULL); end if; end module; end use;

, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5MzY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSUYy

use DocumentTools in Do(%MathContainer93 = NULL); end use;

Valor da Integral

use DocumentTools in module() local f,F,g,G,a,b,q, P, s1; if GetProperty(ComboBox0,value) = "Select dA" then Do(%MathContainer92 = MathML[Export]("Select an area element"));return;end if; try exports(IntOrder); catch : SetProperty(ComboBox0, value, "Select dA"); Do(%MathContainer92 = MathML[Export]("Initialize by re-selecting area element")); Do(%MathContainer94 = NULL); return; end try; Do(G = %MathContainer95); Do(g = %MathContainer97); Do(b = %MathContainer96); Do(a = %MathContainer98); Do(P = %MathContainer93); if nops([P])=0 then Do(%MathContainer94 = MathML[Export]("Please provide an integrand")); return; end if; if nops([G])=0 then Do(%MathContainer94 = MathML[Export]("Please provide G")); return; end if; if nops([g])=0 then Do(%MathContainer94 = MathML[Export]("Please provide g")); return; end if; if nops([b])=0 then Do(%MathContainer94 = MathML[Export]("Please provide b")); return; end if; if nops([a])=0 then Do(%MathContainer94 = MathML[Export]("Please provide a")); return; end if; if IntOrder:-X = 1 then s1 := {x}; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer94 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer94 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer94 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer94 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P,[y=g..G, x=a..b]); elif IntOrder:-X = 2 then s1 := {y}; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer94 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer94 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer94 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer94 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P,[x=g..G, y=a..b]); else return; end if; Do(%MathContainer94 = value(q)); end module; end use;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTEY5

use DocumentTools in Do(%MathContainer95 = NULL); end use;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTEY5

use DocumentTools in Do(%MathContainer96 = NULL); end use;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTEY5

use DocumentTools in Do(%MathContainer97 = NULL); end use;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTEY5

use DocumentTools in Do(%MathContainer98 = NULL); end use;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

use DocumentTools, plots, VectorCalculus in module() local F,g,G,a,b, FF, gg, GG, P, bot, top, lft, rgt, fnt, bck, s2, s1, p1,p3,p4,p5,p6, m, gm, Gm, q1,q2,q3,q4,q5,q6; if GetProperty(ComboBox0,value) = "Select dA" then Do(%Plot13 = display(textplot([0,0,"Please select an area element"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; try exports(IntOrder); catch : SetProperty(ComboBox0,value,"Select dA"); Do(%MathContainer92 = NULL); Do(%Plot13 = display(textplot([0,0,"Initialize by re-selecting an area element"],font=[TIMES,ROMAN,14]),axes=none));return; end try; Do(FF = %MathContainer93); Do(GG = %MathContainer95); Do(gg = %MathContainer97); Do(b = %MathContainer96); Do(a = %MathContainer98); if nops([FF])=0 then Do(%Plot13 = display(textplot([0,0,"Please provide an integrand"],font=[TIMES,ROMAN,14]),axes=none));return;end if; if nops([GG])=0 then Do(%Plot13 = display(textplot([0,0,"Please provide G"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; if nops([gg])=0 then Do(%Plot13 = display(textplot([0,0,"Please provide g"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; if nops([b])= 0 then Do(%Plot13 = display(textplot([0,0,"Please provide b"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; if nops([a])= 0 then Do(%Plot13 = display(textplot([0,0,"Please provide a"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; p1 := display(textplot([0,0," "]),axes=none); p3 := display(textplot([0,0,"Check variables in g"],font=[TIMES,ROMAN,14]),axes=none); p4 := display(textplot([0,0,"Check variables in G"],font=[TIMES,ROMAN,14]),axes=none); p5 := display(textplot([0,0,"Lower limit 'a' must be numeric"],font=[TIMES,ROMAN,14]),axes=none); p6 := display(textplot([0,0,"Upper limit 'b' must be numeric"],font=[TIMES,ROMAN,14]),axes=none); F := unapply(FF,(x,y)); if IntOrder:-X = 1 then s1 := {x}; if nops(select(type,indets(gg),name) minus s1)>0 then Do(%Plot13 = p3);Do(%Plot12 = p1);return;end if; if nops(select(type,indets(GG),name) minus s1)>0 then Do(%Plot13 = p4);Do(%Plot12 = p1);return;end if; if not type(evalf(a),numeric) then Do(%Plot13 = p5);Do(%Plot12 = p1);return;end if; if not type(evalf(b),numeric) then Do(%Plot13 = p6);Do(%Plot12 = p1);return;end if; g := unapply(gg,x); G := unapply(GG,x); bot := plot3d([x,y,0],x=a..b,y=g(x)..G(x),color=blue): top := plot3d([x,y,F(x,y)],x=a..b,y=g(x)..G(x),color=red): lft := plot3d([x,g(x),z],x=a..b,z=0..F(x,g(x)),color=green): rgt := plot3d([x,G(x),z],x=a..b,z=0..F(x,G(x)),color=gold): fnt := plot3d([b,y,z],y=g(b)..G(b),z=0..F(b,y),color=gray): bck := plot3d([a,y,z],y=g(a)..G(a),z=0..F(a,y),color=yellow): elif IntOrder:-X = 2 then s1 := {y}; if nops(select(type,indets(gg),name) minus s1)>0 then Do(%Plot13 = p3);Do(%Plot12 = p1);return;end if; if nops(select(type,indets(GG),name) minus s1)>0 then Do(%Plot13 = p4);Do(%Plot12 = p1);return;end if; if not type(evalf(a),numeric) then Do(%Plot13 = p5);Do(%Plot12 = p1);return;end if; if not type(evalf(b),numeric) then Do(%Plot13 = p6);Do(%Plot12 = p1);return;end if; g := unapply(gg,y); G := unapply(GG,y); bot := plot3d([x,y,0],y=a..b,x=g(y)..G(y),color=blue): top := plot3d([x,y,F(x,y)],y=a..b,x=g(y)..G(y),color=red): lft := plot3d([x,a,z],x=g(a)..G(a),z=0..F(x,a),color=yellow): rgt := plot3d([x,b,z],x=g(b)..G(b),z=0..F(x,b),color=gray): bck := plot3d([g(y),y,z],y=a..b,z=0..F(g(y),y),color=green): fnt := plot3d([G(y),y,z],y=a..b,z=0..F(G(y),y),color=gold): else return; end if; P := display([bot,top,lft,rgt,fnt,bck],axes=box,labels=[x,y,z],orientation=[-115,80],style=surface); Do(%Plot13 = P); m := (a+b)/2; gm := g(m); Gm := G(m); if IntOrder:-X = 1 then q1 := plot(g(x),x=a..b,color=red,legend=typeset(Typesetting:-mi("g"))): q2 := plot(G(x),x=a..b,color=black,legend=typeset(Typesetting:-mi("G"))): q3 := PlotVector(RootedVector(root=[m,gm],[0,Gm-gm]),color="Orange",width=[.05,relative=false]): q4 := plot([[a,g(a)],[a,G(a)]],style=line, color=cyan): q5 := plot([[b,g(b)],[b,G(b)]],style=line, color=cyan): q6 := display([q1,q2,q3,q4,q5],labels=[x,y]); else q1 := plot([g(y),y,y=a..b], color=red,legend=typeset(Typesetting:-mi("g"))): q2 := plot([G(y),y,y=a..b], color=black, legend=typeset(Typesetting:-mi("G"))): q3 := PlotVector(RootedVector(root=[gm,m],[Gm-gm,0]),color="Orange",width=[.05,relative=false]): q4 := plot([[g(a),a],[G(a),a]],style=line, color=cyan): q5 := plot([[g(b),b],[G(b),b]],style=line, color=cyan): q6 := display([q1,q2,q3,q4,q5],labels=[x,y]); end if; Do(%Plot12 = q6); end module; end use;

use DocumentTools, plots in module() local p; p := display(textplot([0,0," "]),axes=none); Do(%Plot12 = p); Do(%Plot13 = p); end module; end use;

use DocumentTools, plots in module() local p; p := display(textplot([0,0," "]),axes=none); Do(%Plot12 = p); Do(%Plot13 = p); SetProperty(ComboBox0, value, "Select dA"); Do(%MathContainer92 = NULL); Do(%MathContainer93 = NULL); Do(%MathContainer95 = NULL); Do(%MathContainer97 = NULL); Do(%MathContainer96 = NULL); Do(%MathContainer98 = NULL); Do(%MathContainer94 = NULL); end module; end use;

Curvas Limitantes

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"Volume"

Tabela 29 Modelo de tarefa para uma integral dupla iterada em coordenadas Cartesianas

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Depois de selecionar a ordem de integra\303\247\303\243o (aqui, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEjZHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSJ+RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtRiw2JVEjZHhGJ0YvRjJGOQ==), e entrando com os dados relevantes, o gr\303\241fico mostra que a primeira integral ( ou interna ) est\303\241 na dire\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. O gr\303\241fico da esquerda mostra a regi\303\243o cujo "volume" \303\251 computado se a fun\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjJGQUZBRkE= \303\251 interpretada como definindo uma superf\303\255cie 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.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 3" layout="Heading 3">Coordenadas Cartesianas: 3D</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Compute o volume no primeiro octante limitado pelo cilindro 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 e o plano LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJ4RidGL0YyRjk=.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

A regi\303\243o cujo volume \303\251 para ser computado est\303\241 mostrada abaixo. O modelo de tarefas na Tabela 30 providencia uma imagem desta regi\303\243o.

Ferramentas\342\211\273Tarefas\342\211\273Pesquisar:

Calculus - Multivariate\342\211\273Integration\342\211\273Visualizing Regions of Integration\342\211\273Cartesian 3-D

Calcule 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 e construa o Gr\303\241fico de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=

Elemento de Volume LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZHZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn

Select dv,dz dy dx,dz dx dy,dx dy dz,dx dz dy,dy dx dz,dy dz dxuse DocumentTools in IntOrder := module() local Y,q1,f,F,g,G,x,y,z,a,b,Psi,q2,q3,q4,q5,q6; export X; q1 := Int(Psi,[z=f(x,y)..F(x,y), y=g(x)..G(x), x=a..b]); q2 := Int(Psi,[z=f(x,y)..F(x,y), x=g(y)..G(y), y=a..b]); q3 := Int(Psi,[x=f(y,z)..F(y,z), y=g(z)..G(z), z=a..b]); q4 := Int(Psi,[x=f(y,z)..F(y,z), z=g(y)..G(y), y=a..b]); q5 := Int(Psi,[y=f(x,z)..F(x,z), x=g(z)..G(z), z=a..b]); q6 := Int(Psi,[y=f(x,z)..F(x,z), z=g(x)..G(x), x=a..b]); Y := GetProperty(ComboBox1,value); if Y = "dz dy dx" then X:=1; Do(%MathContainer99 = q1); elif Y = "dz dx dy" then X:=2; Do(%MathContainer99 = q2); elif Y = "dx dy dz" then X:=3; Do(%MathContainer99 = q3); elif Y = "dx dz dy" then X:=4; Do(%MathContainer99 = q4); elif Y = "dy dx dz" then X:=5; Do(%MathContainer99 = q5); elif Y = "dy dz dx" then X:=6; Do(%MathContainer99 = q6); elif Y = "Select dv" then X:=7; Do(%MathContainer99 = NULL); end if; end module; end use;

, onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5MzY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSUYy

use DocumentTools in Do(%MathContainer100 = NULL); end use;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTEY5

use DocumentTools in Do(%MathContainer101 = NULL); end use;

use DocumentTools in Do(%MathContainer102 = NULL); end use;

use DocumentTools in Do(%MathContainer103 = NULL); end use;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTEY5

use DocumentTools in Do(%MathContainer104 = NULL); end use;

use DocumentTools in Do(%MathContainer105 = NULL); end use;

use DocumentTools in Do(%MathContainer106 = NULL); end use;

use DocumentTools in module() local f,F,g,G,a,b,q, P, s1,s2; if GetProperty(ComboBox1,value) = "Select dv" then Do(%MathContainer99 = MathML[Export]("Select a volume element"));return;end if; try exports(IntOrder); catch : SetProperty(ComboBox1, value, "Select dv"); Do(%MathContainer99 = MathML[Export]("Initialize by re-selecting volume element")); Do(%MathContainer107 = NULL); return; end try; Do(F = %MathContainer101); Do(f = %MathContainer104); Do(G = %MathContainer102); Do(g = %MathContainer105); Do(b = %MathContainer103); Do(a = %MathContainer106); Do(P = %MathContainer100); if nops([P])=0 then Do(%MathContainer107 = MathML[Export]("Please provide an integrand")); return; end if; if nops([F])=0 then Do(%MathContainer107 = MathML[Export]("Please provide F")); return; end if; if nops([f])=0 then Do(%MathContainer107 = MathML[Export]("Please provide f")); return; end if; if nops([G])=0 then Do(%MathContainer107 = MathML[Export]("Please provide G")); return; end if; if nops([g])=0 then Do(%MathContainer107 = MathML[Export]("Please provide g")); return; end if; if nops([b])=0 then Do(%MathContainer107 = MathML[Export]("Please provide b")); return; end if; if nops([a])=0 then Do(%MathContainer107 = MathML[Export]("Please provide a")); return; end if; if IntOrder:-X = 1 then s2 := {x,y}; s1 := {x}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer107 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer107 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer107 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer107 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer107 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer107 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P,[z=f..F, y=g..G, x=a..b]); elif IntOrder:-X = 2 then s2 := {x,y}; s1 := {y}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer107 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer107 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer107 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer107 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer107 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer107 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P,[z=f..F, x=g..G, y=a..b]); elif IntOrder:-X = 3 then s2 := {y,z}; s1 := {z}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer107 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer107 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer107 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer107 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer107 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer107 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P,[x=f..F, y=g..G, z=a..b]); elif IntOrder:-X = 4 then s2 := {y,z}; s1 := {y}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer107 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer107 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer107 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer107 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer107 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer107 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P,[x=f..F, z=g..G, y=a..b]); elif IntOrder:-X = 5 then s2 := {x,z}; s1 := {z}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer107 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer107 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer107 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer107 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer107 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer107 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P,[y=f..F, x=g..G, z=a..b]); elif IntOrder:-X = 6 then s2 := {x,z}; s1 := {x}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer107 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer107 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer107 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer107 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer107 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer107 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P,[y=f..F, z=g..G, x=a..b]); else return; end if; Do(%MathContainer107 = value(q)); end module; end use;

use DocumentTools in module() local f,F,g,G,a,b,q, P; if GetProperty(ComboBox1,value) = "Select dv" then Do(%MathContainer99 = MathML[Export]("Select a volume element"));return;end if; try exports(IntOrder); catch : SetProperty(ComboBox1, value, "Select dv"); Do(%MathContainer99 = MathML[Export]("Initialize by re-selecting volume element")); Do(%MathContainer107 = NULL); return; end try; Do(F = %MathContainer101); Do(f = %MathContainer104); Do(G = %MathContainer102); Do(g = %MathContainer105); Do(b = %MathContainer103); Do(a = %MathContainer106); Do(P = %MathContainer100); if nops([F])=0 then Do(%MathContainer107 = MathML[Export]("Please provide F")); return; end if; if nops([f])=0 then Do(%MathContainer107 = MathML[Export]("Please provide f")); return; end if; if nops([G])=0 then Do(%MathContainer107 = MathML[Export]("Please provide G")); return; end if; if nops([g])=0 then Do(%MathContainer107 = MathML[Export]("Please provide g")); return; end if; if nops([b])=0 then Do(%MathContainer107 = MathML[Export]("Please provide b")); return; end if; if nops([a])=0 then Do(%MathContainer107 = MathML[Export]("Please provide a")); return; end if; if nops([P])=0 then Do(%MathContainer107 = MathML[Export]("Please provide an integrand")); return; end if; if IntOrder:-X = 1 then s2 := {x,y}; s1 := {x}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer107 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer107 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer107 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer107 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer107 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer107 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P,[z=f..F, y=g..G, x=a..b]); elif IntOrder:-X = 2 then s2 := {x,y}; s1 := {y}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer107 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer107 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer107 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer107 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer107 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer107 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P,[z=f..F, x=g..G, y=a..b]); elif IntOrder:-X = 3 then s2 := {y,z}; s1 := {z}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer107 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer107 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer107 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer107 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer107 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer107 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P,[x=f..F, y=g..G, z=a..b]); elif IntOrder:-X = 4 then s2 := {y,z}; s1 := {y}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer107 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer107 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer107 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer107 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer107 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer107 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P,[x=f..F, z=g..G, y=a..b]); elif IntOrder:-X = 5 then s2 := {x,z}; s1 := {z}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer107 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer107 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer107 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer107 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer107 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer107 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P,[y=f..F, x=g..G, z=a..b]); elif IntOrder:-X = 6 then s2 := {x,z}; s1 := {x}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer107 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer107 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer107 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer107 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer107 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer107 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P,[y=f..F, z=g..G, x=a..b]); else return; end if; Do(%MathContainer107 = evalf(q)); end module; end use;

use DocumentTools, plots in module() local f,F,g,G,a,b, ff, FF, gg, GG, P, bot, top, lft, rgt, fnt, bck, s2, s1, p1,p2,p3,p4,p5,p6; if GetProperty(ComboBox1,value) = "Select dv" then Do(%Plot14 = display(textplot([0,0,"Please select a volume element"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; try exports(IntOrder); catch : SetProperty(ComboBox1,value,"Select dv"); Do(%MathContainer99 = NULL); Do(%Plot14 = display(textplot([0,0,"Initialize by re-selecting a volume element"],font=[TIMES,ROMAN,14]),axes=none));return; end try; Do(FF = %MathContainer101); Do(ff = %MathContainer104); Do(GG = %MathContainer102); Do(gg = %MathContainer105); Do(b = %MathContainer103); Do(a = %MathContainer106); if nops([FF])=0 then Do(%Plot14 = display(textplot([0,0,"Please provide F"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; if nops([ff])=0 then Do(%Plot14 = display(textplot([0,0,"Please provide f"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; if nops([GG])=0 then Do(%Plot14 = display(textplot([0,0,"Please provide G"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; if nops([gg])=0 then Do(%Plot14 = display(textplot([0,0,"Please provide g"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; if nops([b])= 0 then Do(%Plot14 = display(textplot([0,0,"Please provide b"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; if nops([a])= 0 then Do(%Plot14 = display(textplot([0,0,"Please provide a"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; p1 := display(textplot([0,0,"Check variables in f"],font=[TIMES,ROMAN,14]),axes=none); p2 := display(textplot([0,0,"Check variables in F"],font=[TIMES,ROMAN,14]),axes=none); p3 := display(textplot([0,0,"Check variables in g"],font=[TIMES,ROMAN,14]),axes=none); p4 := display(textplot([0,0,"Check variables in G"],font=[TIMES,ROMAN,14]),axes=none); p5 := display(textplot([0,0,"Lower limit 'a' must be numeric"],font=[TIMES,ROMAN,14]),axes=none); p6 := display(textplot([0,0,"Upper limit 'b' must be numeric"],font=[TIMES,ROMAN,14]),axes=none); if IntOrder:-X = 1 then s2 := {x,y}; s1 := {x}; if nops(select(type,indets(ff),name) minus s2)>0 then Do(%MathContainer107 = p1);return;end if; if nops(select(type,indets(FF),name) minus s2)>0 then Do(%MathContainer107 = p2);return;end if; if nops(select(type,indets(gg),name) minus s1)>0 then Do(%MathContainer107 = p3);return;end if; if nops(select(type,indets(GG),name) minus s1)>0 then Do(%MathContainer107 = p4);return;end if; if not type(evalf(a),numeric) then Do(%MathContainer107 = p5);return;end if; if not type(evalf(b),numeric) then Do(%MathContainer107 = p6);return;end if; f := unapply(ff,(x,y)); F := unapply(FF,(x,y)); g := unapply(gg,x); G := unapply(GG,x); bot := plot3d([x,y,f(x,y)],x=a..b,y=g(x)..G(x),color=blue): top := plot3d([x,y,F(x,y)],x=a..b,y=g(x)..G(x),color=red): lft := plot3d([x,g(x),z],x=a..b,z=f(x,g(x))..F(x,g(x)),color=green): rgt := plot3d([x,G(x),z],x=a..b,z=f(x,G(x))..F(x,G(x)),color=gold): fnt := plot3d([b,y,z],y=g(b)..G(b),z=f(b,y)..F(b,y),color=gray): bck := plot3d([a,y,z],y=g(a)..G(a),z=f(a,y)..F(a,y),color=yellow): elif IntOrder:-X = 2 then s2 := {x,y}; s1 := {y}; if nops(select(type,indets(ff),name) minus s2)>0 then Do(%MathContainer107 = p1);return;end if; if nops(select(type,indets(FF),name) minus s2)>0 then Do(%MathContainer107 = p2);return;end if; if nops(select(type,indets(gg),name) minus s1)>0 then Do(%MathContainer107 = p3);return;end if; if nops(select(type,indets(GG),name) minus s1)>0 then Do(%MathContainer107 = p4);return;end if; if not type(evalf(a),numeric) then Do(%MathContainer107 = p5);return;end if; if not type(evalf(b),numeric) then Do(%MathContainer107 = p6);return;end if; f := unapply(ff,(x,y)); F := unapply(FF,(x,y)); g := unapply(gg,y); G := unapply(GG,y); bot := plot3d([x,y,f(x,y)],y=a..b,x=g(y)..G(y),color=blue): top := plot3d([x,y,F(x,y)],y=a..b,x=g(y)..G(y),color=red): lft := plot3d([x,a,z],x=g(a)..G(a),z=f(x,a)..F(x,a),color=yellow): rgt := plot3d([x,b,z],x=g(b)..G(b),z=f(x,b)..F(x,b),color=gray): bck := plot3d([g(y),y,z],y=a..b,z=f(g(y),y)..F(g(y),y),color=green): fnt := plot3d([G(y),y,z],y=a..b,z=f(G(y),y)..F(G(y),y),color=gold): elif IntOrder:-X = 3 then s2 := {y,z}; s1 := {z}; if nops(select(type,indets(ff),name) minus s2)>0 then Do(%MathContainer107 = p1);return;end if; if nops(select(type,indets(FF),name) minus s2)>0 then Do(%MathContainer107 = p2);return;end if; if nops(select(type,indets(gg),name) minus s1)>0 then Do(%MathContainer107 = p3);return;end if; if nops(select(type,indets(GG),name) minus s1)>0 then Do(%MathContainer107 = p4);return;end if; if not type(evalf(a),numeric) then Do(%MathContainer107 = p5);return;end if; if not type(evalf(b),numeric) then Do(%MathContainer107 = p6);return;end if; f := unapply(ff,(y,z)); F := unapply(FF,(y,z)); g := unapply(gg,z); G := unapply(GG,z); bot := plot3d([x,y,a],x=f(y,a)..F(y,a),y=g(a)..G(a),color=yellow): top := plot3d([x,y,b],x=f(y,b)..F(y,b),y=g(b)..G(b),color=gray): lft := plot3d([x,g(z),z],z=a..b,x=f(g(z),z)..F(g(z),z),color=green): rgt := plot3d([x,G(z),z],z=a..b,x=f(G(z),z)..F(G(z),z),color=gold): fnt := plot3d([f(y,z),y,z],z=a..b,y=g(z)..G(z),color=blue): bck := plot3d([F(y,z),y,z],z=a..b,y=g(z)..G(z),color=red): elif IntOrder:-X = 4 then s2 := {y,z}; s1 := {y}; if nops(select(type,indets(ff),name) minus s2)>0 then Do(%MathContainer107 = p1);return;end if; if nops(select(type,indets(FF),name) minus s2)>0 then Do(%MathContainer107 = p2);return;end if; if nops(select(type,indets(gg),name) minus s1)>0 then Do(%MathContainer107 = p3);return;end if; if nops(select(type,indets(GG),name) minus s1)>0 then Do(%MathContainer107 = p4);return;end if; if not type(evalf(a),numeric) then Do(%MathContainer107 = p5);return;end if; if not type(evalf(b),numeric) then Do(%MathContainer107 = p6);return;end if; f := unapply(ff,(y,z)); F := unapply(FF,(y,z)); g := unapply(gg,y); G := unapply(GG,y); bot := plot3d([x,y,g(y)],y=a..b,x=f(y,g(y))..F(y,g(y)),color=green): top := plot3d([x,y,G(y)],y=a..b,x=f(y,G(y))..F(y,G(y)),color=gold): lft := plot3d([x,a,z],x=f(a,z)..F(a,z),z=g(a)..G(a),color=yellow): rgt := plot3d([x,b,z],x=f(b,z)..F(b,z),z=g(b)..G(b),color=gray): fnt := plot3d([F(y,z),y,z],y=a..b,z=g(y)..G(y),color=red): bck := plot3d([f(y,z),y,z],y=a..b,z=g(y)..G(y),color=blue): elif IntOrder:-X = 5 then s2 := {x,z}; s1 := {z}; if nops(select(type,indets(ff),name) minus s2)>0 then Do(%MathContainer107 = p1);return;end if; if nops(select(type,indets(FF),name) minus s2)>0 then Do(%MathContainer107 = p2);return;end if; if nops(select(type,indets(gg),name) minus s1)>0 then Do(%MathContainer107 = p3);return;end if; if nops(select(type,indets(GG),name) minus s1)>0 then Do(%MathContainer107 = p4);return;end if; if not type(evalf(a),numeric) then Do(%MathContainer107 = p5);return;end if; if not type(evalf(b),numeric) then Do(%MathContainer107 = p6);return;end if; f := unapply(ff,(x,z)); F := unapply(FF,(x,z)); g := unapply(gg,z); G := unapply(GG,z); bot := plot3d([x,y,a],y=f(x,a)..F(x,a),x=g(a)..G(a),color=yellow): top := plot3d([x,y,b],y=f(x,b)..F(x,b),x=g(b)..G(b),color=gray): lft := plot3d([x,f(x,z),z],z=a..b,x=g(z)..G(z),color=blue): rgt := plot3d([x,F(x,z),z],z=a..b,x=g(z)..G(z),color=red): fnt := plot3d([G(z),y,z],z=a..b,y=f(G(z),z)..F(G(z),z),color=gold): bck := plot3d([g(z),y,z],z=a..b,y=f(g(z),z)..F(g(z),z),color=green): elif IntOrder:-X = 6 then s2 := {x,z}; s1 := {x}; if nops(select(type,indets(ff),name) minus s2)>0 then Do(%MathContainer107 = p1);return;end if; if nops(select(type,indets(FF),name) minus s2)>0 then Do(%MathContainer107 = p2);return;end if; if nops(select(type,indets(gg),name) minus s1)>0 then Do(%MathContainer107 = p3);return;end if; if nops(select(type,indets(GG),name) minus s1)>0 then Do(%MathContainer107 = p4);return;end if; if not type(evalf(a),numeric) then Do(%MathContainer107 = p5);return;end if; if not type(evalf(b),numeric) then Do(%MathContainer107 = p6);return;end if; f := unapply(ff,(x,z)); F := unapply(FF,(x,z)); g := unapply(gg,x); G := unapply(GG,x); bot := plot3d([x,y,g(x)],x=a..b,y=f(x,g(x))..F(x,g(x)),color=green): top := plot3d([x,y,G(x)],x=a..b,y=f(x,G(x))..F(x,G(x)),color=gold): lft := plot3d([x,f(x,z),z],x=a..b,z=g(x)..G(x),color=blue): rgt := plot3d([x,F(x,z),z],x=a..b,z=g(x)..G(x),color=red): fnt := plot3d([b,y,z],y=f(x,z)..F(b,z),z=g(b)..G(b),color=gray): bck := plot3d([a,y,z],y=f(x,z)..F(a,z),z=g(a)..G(a),color=yellow): else return; end if; P := display([bot,top,lft,rgt,fnt,bck],axes=box,labels=[x,y,z],orientation=[-115,80],style=surface); Do(%Plot14 = P); end module; end use;

use DocumentTools, plots in Do(%Plot14 = display(textplot([0,0," "]),axes=none)); end use;

use DocumentTools, plots in Do(%MathContainer99 = NULL); Do(%MathContainer100 = NULL); Do(%MathContainer101 = NULL); Do(%MathContainer104 = NULL); Do(%MathContainer102 = NULL); Do(%MathContainer105 = NULL); Do(%MathContainer103 = NULL); Do(%MathContainer106 = NULL); Do(%MathContainer107 = NULL); Do(%Plot14 = display(textplot([0,0," "]),axes=none)); SetProperty(ComboBox1, value, "Select Order"); end use;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Tabela 30 Modelo de tarefa para uma integral tripla iterada em coordenadas Cartesianas

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Uma ordem comum para uma integral tripla iterada em coordenadas cartesianas \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEjZHpGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSJ+RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtRiw2JVEjZHlGJ0YvRjJGNS1GLDYlUSNkeEYnRi9GMkY5. Surpreendentemente, a ordem LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEjZHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSJ+RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtRiw2JVEjZHlGJ0YvRjJGNS1GLDYlUSNkekYnRi9GMkY5 usada na Tabela 30 resulta em uma integral iterada cujo limite superior em ambos os casos \303\251 o mesmo! O feedback visual do gr\303\241fico na Tabela 30 sugere que a regi\303\243o foi corretamente varrida escrevendo aintegral como mostrado.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 3" layout="Heading 3">Coordenadas Polares</Text-field>

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Calcule a \303\241rea limitada pela curva 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.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

O modelo de tarefas na Tabela 31 providencia uma solu\303\247\303\243o. A localiza\303\247\303\243o do modelo de tarefas \303\251 dada no topo da tabela

Ferramentas\342\211\273Tarefas\342\211\273Pesquisar:

Calculus - Multivariate\342\211\273Integration\342\211\273Visualizing Regions of Integration\342\211\273Polar

Calcule 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 e fa\303\247a o Gr\303\241fico de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=

Elemento de \303\200rea LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JlEjZEFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjZRJ25vcm1hbEYn

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSNkckYnRi9GMkY1LUYsNiVRKGQmIzk1MjtGJ0YvRjJGNUY5

use DocumentTools in module() local Q, r,theta,x,y,Psi,g,G,a,b; if Do(%RadioButton8) then Q:=Int(Psi*r,[r=g(theta)..G(theta),theta=a..b]); else Q:=Int(Psi*r,[theta=g(r)..G(r),r=a..b]);end if; Do(%MathContainer108 = Q); end module; end use;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUShkJiM5NTI7RidGL0YyRjUtRiw2JVEjZHJGJ0YvRjJGNS8lK2V4ZWN1dGFibGVHRj1GOQ==

KihJInJHNiIiIiJJKWQmdGhldGE7R0YkRiVJI2RyR0YkRiU=

use DocumentTools in module() local Q,r,theta,x,y,Psi,g,G,a,b; if Do(%RadioButton9) then Q:=Int(Psi*r,[theta=g(r)..G(r),r=a..b]); else Q:=Int(Psi*r,[r=g(theta)..G(theta),theta=a..b]); end if; Do(%MathContainer108 = Q); end module; end use;

use DocumentTools in Do(%MathContainer109 = NULL); end use;

Valor da Integral

use DocumentTools in module() local f,F,g,G,a,b,q, P, s1; if Do(%RadioButton10) then Do(%MathContainer108 = MathML[Export]("Select an area element"));return;end if; Do(G = %MathContainer111); Do(g = %MathContainer113); Do(b = %MathContainer112); Do(a = %MathContainer114); Do(P = %MathContainer109); if nops([P])=0 then Do(%MathContainer110 = MathML[Export]("Please provide an integrand")); return; end if; if nops([G])=0 then Do(%MathContainer110 = MathML[Export]("Please provide G")); return; end if; if nops([g])=0 then Do(%MathContainer110 = MathML[Export]("Please provide g")); return; end if; if nops([b])=0 then Do(%MathContainer110 = MathML[Export]("Please provide b")); return; end if; if nops([a])=0 then Do(%MathContainer110 = MathML[Export]("Please provide a")); return; end if; if Do(%RadioButton8) then s1 := {theta}; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer110 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer110 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer110 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer110 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P*r,[r=g..G, theta=a..b]); elif Do(%RadioButton9) then s1 := {r}; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer110 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer110 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer110 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer110 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P*r,[theta=g..G, r=a..b]); else return; end if; Do(%MathContainer110 = value(q)); end module; end use;

use DocumentTools in Do(%MathContainer111 = NULL); end use;

use DocumentTools in Do(%MathContainer112 = NULL); end use;

use DocumentTools in Do(%MathContainer113 = NULL); end use;

use DocumentTools in Do(%MathContainer114 = NULL); end use;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

use DocumentTools, plots in module() local F,g,G,a,b, FF, gg, GG, P, bot, top, lft, rgt, fnt, bck, s1, p1,p3,p4,p5,p6, q1,q2,q3,q4,q6, k; if Do(%RadioButton10) then Do(%Plot16 = display(textplot([0,0,"Please select an area element"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; Do(FF = %MathContainer109); Do(GG = %MathContainer111); Do(gg = %MathContainer113); Do(b = %MathContainer112); Do(a = %MathContainer114); if nops([FF])=0 then Do(%Plot16 = display(textplot([0,0,"Please provide an integrand"],font=[TIMES,ROMAN,14]),axes=none));return;end if; if nops([GG])=0 then Do(%Plot16 = display(textplot([0,0,"Please provide G"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; if nops([gg])=0 then Do(%Plot16 = display(textplot([0,0,"Please provide g"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; if nops([b])= 0 then Do(%Plot16 = display(textplot([0,0,"Please provide b"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; if nops([a])= 0 then Do(%Plot16 = display(textplot([0,0,"Please provide a"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; p1 := display(textplot([0,0," "]),axes=none); p3 := display(textplot([0,0,"Check variables in g"],font=[TIMES,ROMAN,14]),axes=none); p4 := display(textplot([0,0,"Check variables in G"],font=[TIMES,ROMAN,14]),axes=none); p5 := display(textplot([0,0,"Lower limit 'a' must be numeric"],font=[TIMES,ROMAN,14]),axes=none); p6 := display(textplot([0,0,"Upper limit 'b' must be numeric"],font=[TIMES,ROMAN,14]),axes=none); F := unapply(FF,(r,theta)); if Do(%RadioButton8) then s1 := {theta}; if nops(select(type,indets(gg),name) minus s1)>0 then Do(%Plot16 = p3);Do(%Plot15 = p1);return;end if; if nops(select(type,indets(GG),name) minus s1)>0 then Do(%Plot16 = p4);Do(%Plot15 = p1);return;end if; if not type(evalf(a),numeric) then Do(%Plot16 = p5);Do(%Plot15 = p1);return;end if; if not type(evalf(b),numeric) then Do(%Plot16 = p6);Do(%Plot15 = p1);return;end if; g := unapply(gg,theta); G := unapply(GG,theta); top := plot3d([r,t,F(r,t)],r=g(t)..G(t),t=a..b,coords=cylindrical): bot := plot3d([r,t,0], r=g(t)..G(t),t=a..b,coords=cylindrical): bck := plot3d([g(t),t,z],t=a..b,z=0..F(g(t),t),coords=cylindrical): fnt := plot3d([G(t),t,z],t=a..b,z=0..F(G(t),t),coords=cylindrical): lft := plot3d([r,a,z],r=g(a)..G(a),z=0..F(r,a),coords=cylindrical): rgt := plot3d([r,b,z],r=g(b)..G(b),z=0..F(r,b),coords=cylindrical): elif Do(%RadioButton9) then s1 := {r}; if nops(select(type,indets(gg),name) minus s1)>0 then Do(%Plot16 = p3);Do(%Plot15 = p1);return;end if; if nops(select(type,indets(GG),name) minus s1)>0 then Do(%Plot16 = p4);Do(%Plot15 = p1);return;end if; if not type(evalf(a),numeric) then Do(%Plot16 = p5);Do(%Plot15 = p1);return;end if; if not type(evalf(b),numeric) then Do(%Plot16 = p6);Do(%Plot15 = p1);return;end if; g := unapply(gg,r); G := unapply(GG,r); top := plot3d([r,t,F(r,t)],t=g(r)..G(r),r=a..b,coords=cylindrical): bot := plot3d([r,t,0], t=g(r)..G(r),r=a..b,coords=cylindrical): bck := plot3d([a,t,z],t=g(a)..G(a),z=0..F(a,t),coords=cylindrical): fnt := plot3d([b,t,z],t=g(b)..G(b),z=0..F(b,t),coords=cylindrical): lft := plot3d([r,g(r),z],r=a..b,z=0..F(r,g(r)),coords=cylindrical): rgt := plot3d([r,G(r),z],r=a..b,z=0..F(r,G(r)),coords=cylindrical): else return; end if; P := display([bot,top,lft,rgt,fnt,bck],axes=box,labels=[x,y,z],orientation=[-115,80]); Do(%Plot16 = P); if Do(%RadioButton8) then q1 := plot([[g(t),t,t=a..b],[G(t),t,t=a..b],[r,a,r=g(a)..G(a)],[r,b,r=g(b)..G(b)]],coords=polar, color=[red,green,blue,blue]): q2 := animate(plot, [[k*g(theta)+(1-k)*G(theta),theta,k=0..1],coords=polar,color="Orange",thickness=3],theta=a..b,frames=25): q3 := animate(plot,[[[g(theta),theta]],style=point,coords=polar, symbol=circle,color=black,symbolsize=20],theta=a..b,frames=25): q4 := animate(plot,[[[G(theta),theta]],style=point,coords=polar, symbol=solidcircle,color=black,symbolsize=20],theta=a..b,frames=25): q6 := display([q1,q2,q3,q4],scaling=constrained,labels=[x,y]); else q1 := plot([[r,g(r),r=a..b],[r,G(r),r=a..b],[a,t,t=g(a)..G(a)],[b,t,t=g(b)..G(b)]], coords=polar, color=[red,green,blue,blue]): q2 := animate(plot, [[r,k*g(r)+(1-k)*G(r),k=0..1],coords=polar,color="Orange",thickness=3],r=a..b,frames=25): q3 := animate(plot,[[[r,g(r)]],style=point,coords=polar, symbol=circle,color=black,symbolsize=20],r=a..b,frames=25): q4 := animate(plot,[[[r,G(r)]],style=point,coords=polar, symbol=solidcircle,color=black,symbolsize=20],r=a..b,frames=25): q6 := display([q1,q2,q3,q4],scaling=constrained,labels=[x,y]); end if; Do(%Plot15 = q6); end module; end use;

use DocumentTools, plots in module() local p; p := display(textplot([0,0," "]),axes=none); Do(%Plot15 = p); Do(%Plot16 = p); end module; end use;

use DocumentTools, plots in module() local p; p := display(textplot([0,0," "]),axes=none); Do(%Plot15 = p); Do(%Plot16 = p); Do(%MathContainer108 = NULL); Do(%MathContainer109 = NULL); Do(%MathContainer111 = NULL); Do(%MathContainer113 = NULL); Do(%MathContainer112 = NULL); Do(%MathContainer114 = NULL); Do(%MathContainer110 = NULL); Do(%RadioButton10(value)=true); end module; end use;

Curvas Limitantes

"Volume"

Tabela 31 Modelo de tarefas para uma integral dupla iterada em coordenadas polares

O gr\303\241fico na esquerda \303\251 uma anima\303\247\303\243o, mostrando como o raio varia com o \303\242ngulo. O gr\303\241fico na direita mostra o volume computado quando o integrando \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5MzY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjFGJ0YyRjI=. O n\303\272mero computado pela integral \303\251 ent\303\243o a \303\241rea requisitada.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 3" layout="Heading 3">Coordenadas Cil\303\255ndricas</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Use coordenadas cil\303\255ndricas para calcular o volume do s\303\263lido tirado de uma esfera unit\303\241ria cortada por um cone cujo v\303\251rtice est\303\241 no centro da esfera e cujo gerador faz um \303\242ngulo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEjMzBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21vR0YkNi1RKSZjb21wZm47RidGMi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjE2NjY2NjdlbUYnLyUncnNwYWNlR0ZMLyUnaXRhbGljR1EldHJ1ZUYnL0YzUSdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGMg==com o eixo de simetria do cone.

O modelo de tarefa na Tabela 32 providencia uma solu\303\247\303\243o. A localiza\303\247\303\243o do modelo de tarefas \303\251 dada no topo da tabela.

Ferramentas\342\211\273Tarefas\342\211\273Pesquisar:

Calculus - Multivariate\342\211\273Integration\342\211\273Visualizing Regions of Integration\342\211\273Cylindrical

Calcule 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 e fa\303\247a o Gr\303\241fico de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=

Elemento de Volume LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZHZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSNkekYnRi9GMkY1LUYsNiVRI2RyRidGL0YyRjUtRiw2JVEiZEYnRi9GMi1GLDYlUScmIzk1MjtGJy9GMEY9RjlGOQ==

use DocumentTools in module() local f,F,g,G,Psi,theta,r,x,y,z,a,b; Do(%MathContainer115 = Int(Psi*r,[z=f(r,theta)..F(r,theta), r=g(theta)..G(theta), theta=a..b])); end module; end use;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSNkekYnRi9GMkY1LUYsNiVRImRGJ0YvRjItRiw2JVEnJiM5NTI7RicvRjBGPUY5RjUtRiw2JVEjZHJGJ0YvRjJGOQ==

use DocumentTools in module() local f,F,g,G,Psi,theta,r,x,y,z,a,b; Do(%MathContainer115 = Int(Psi*r,[z=f(r,theta)..F(r,theta), theta=g(r)..G(r), r=a..b])); end module; end use;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSNkckYnRi9GMkY1LUYsNiVRImRGJ0YvRjItRiw2JVEnJiM5NTI7RicvRjBGPUY5RjUtRiw2JVEjZHpGJ0YvRjJGOQ==

use DocumentTools in module() local f,F,g,G,Psi,theta,r,x,y,z,a,b; Do(%MathContainer115 = Int(Psi*r,[r=f(theta,z)..F(theta,z), theta=g(z)..G(z), z=a..b])); end module; end use;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSNkckYnRi9GMkY1LUYsNiVRI2R6RidGL0YyRjUtRiw2JVEiZEYnRi9GMi1GLDYlUScmIzk1MjtGJy9GMEY9RjlGOQ==

use DocumentTools in module() local f,F,g,G,Psi,theta,r,x,y,z,a,b; Do(%MathContainer115 = Int(Psi*r,[r=f(theta,z)..F(theta,z), z=g(theta)..G(theta), theta=a..b])); end module; end use;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJkRidGL0YyLUYsNiVRJyYjOTUyO0YnL0YwRj1GOUY1LUYsNiVRI2RyRidGL0YyRjUtRiw2JVEjZHpGJ0YvRjJGOQ==

use DocumentTools in module() local f,F,g,G,Psi,theta,r,x,y,z,a,b; Do(%MathContainer115 = Int(Psi*r,[theta=f(r,z)..F(r,z), r=g(z)..G(z), z=a..b])); end module; end use;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJkRidGL0YyLUYsNiVRJyYjOTUyO0YnL0YwRj1GOUY1LUYsNiVRI2R6RidGL0YyRjUtRiw2JVEjZHJGJ0YvRjJGOQ==

use DocumentTools in module() local f,F,g,G,Psi,theta,r,x,y,z,a,b; Do(%MathContainer115 = Int(Psi*r,[theta=f(r,z)..F(r,z), z=g(r)..G(r), r=a..b])); end module; end use;

use DocumentTools in Do(%MathContainer116 = NULL); end use;

use DocumentTools in Do(%MathContainer117 = NULL); end use;

use DocumentTools in Do(%MathContainer118 = NULL); end use;

use DocumentTools in Do(%MathContainer119 = NULL); end use;

use DocumentTools in Do(%MathContainer120 = NULL); end use;

use DocumentTools in Do(%MathContainer121 = NULL); end use;

use DocumentTools in Do(%MathContainer122 = NULL); end use;

use DocumentTools in module() local f,F,g,G,a,b,q, P, s1,s2; if Do(%RadioButton11) then Do(%MathContainer115 = MathML[Export]("Select a volume element"));return;end if; Do(F = %MathContainer117); Do(f = %MathContainer120); Do(G = %MathContainer118); Do(g = %MathContainer121); Do(b = %MathContainer119); Do(a = %MathContainer122); Do(P = %MathContainer116); if nops([P])=0 then Do(%MathContainer123 = MathML[Export]("Please provide an integrand")); return; end if; if nops([F])=0 then Do(%MathContainer123 = MathML[Export]("Please provide F")); return; end if; if nops([f])=0 then Do(%MathContainer123 = MathML[Export]("Please provide f")); return; end if; if nops([G])=0 then Do(%MathContainer123 = MathML[Export]("Please provide G")); return; end if; if nops([g])=0 then Do(%MathContainer123 = MathML[Export]("Please provide g")); return; end if; if nops([b])=0 then Do(%MathContainer123 = MathML[Export]("Please provide b")); return; end if; if nops([a])=0 then Do(%MathContainer123 = MathML[Export]("Please provide a")); return; end if; if Do(%RadioButton12) then s2 := {r,theta}; s1 := {theta}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer123 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer123 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer123 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer123 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer123 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer123 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P*r,[z=f..F, r=g..G, theta=a..b]); elif Do(%RadioButton13) then s2 := {r,theta}; s1 := {r}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer123 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer123 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer123 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer123 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer123 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer123 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P*r,[z=f..F, theta=g..G, r=a..b]); elif Do(%RadioButton14) then s2 := {theta,z}; s1 := {z}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer123 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer123 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer123 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer123 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer123 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer123 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P*r,[r=f..F, theta=g..G, z=a..b]); elif Do(%RadioButton15) then s2 := {theta,z}; s1 := {theta}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer123 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer123 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer123 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer123 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer123 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer123 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P*r,[r=f..F, z=g..G, theta=a..b]); elif Do(%RadioButton16) then s2 := {r,z}; s1 := {z}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer123 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer123 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer123 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer123 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer123 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer123 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P*r,[theta=f..F, r=g..G, z=a..b]); elif Do(%RadioButton17) then s2 := {r,z}; s1 := {r}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer123 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer123 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer123 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer123 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer123 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer123 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P*r,[theta=f..F, z=g..G, r=a..b]); else return; end if; Do(%MathContainer123 = value(q)); end module; end use;

use DocumentTools in module() local f,F,g,G,a,b,q, P; if Do(%RadioButton11) then Do(%MathContainer115 = MathML[Export]("Select a volume element"));return;end if; Do(F = %MathContainer117); Do(f = %MathContainer120); Do(G = %MathContainer118); Do(g = %MathContainer121); Do(b = %MathContainer119); Do(a = %MathContainer122); Do(P = %MathContainer116); if nops([F])=0 then Do(%MathContainer123 = MathML[Export]("Please provide F")); return; end if; if nops([f])=0 then Do(%MathContainer123 = MathML[Export]("Please provide f")); return; end if; if nops([G])=0 then Do(%MathContainer123 = MathML[Export]("Please provide G")); return; end if; if nops([g])=0 then Do(%MathContainer123 = MathML[Export]("Please provide g")); return; end if; if nops([b])=0 then Do(%MathContainer123 = MathML[Export]("Please provide b")); return; end if; if nops([a])=0 then Do(%MathContainer123 = MathML[Export]("Please provide a")); return; end if; if nops([P])=0 then Do(%MathContainer123 = MathML[Export]("Please provide an integrand")); return; end if; if Do(%RadioButton12) then s2 := {r,theta}; s1 := {theta}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer123 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer123 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer123 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer123 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer123 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer123 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P*r,[z=f..F, r=g..G, theta=a..b]); elif Do(%RadioButton13) then s2 := {r,theta}; s1 := {r}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer123 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer123 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer123 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer123 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer123 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer123 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P*r,[z=f..F, theta=g..G, r=a..b]); elif Do(%RadioButton14) then s2 := {theta,z}; s1 := {z}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer123 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer123 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer123 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer123 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer123 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer123 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P*r,[r=f..F, theta=g..G, z=a..b]); elif Do(%RadioButton15) then s2 := {theta,z}; s1 := {theta}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer123 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer123 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer123 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer123 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer123 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer123 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P*r,[r=f..F, z=g..G, theta=a..b]); elif Do(%RadioButton16) then s2 := {r,z}; s1 := {z}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer123 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer123 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer123 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer123 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer123 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer123 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P*r,[theta=f..F, r=g..G, z=a..b]); elif Do(%RadioButton17) then s2 := {r,z}; s1 := {r}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer123 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer123 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer123 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer123 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer123 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer123 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P*r,[theta=f..F, z=g..G, r=a..b]); else return; end if; Do(%MathContainer123 = evalf(q)); end module; end use;

use DocumentTools, plots in module() local f,F,g,G,a,b, ff, FF, gg, GG, P, S1,S2,S3,S4,S5,S6, s2, s1, p1,p2,p3,p4,p5,p6; if Do(%RadioButton11) then Do(%Plot17 = display(textplot([0,0,"Please select a volume element"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; Do(FF = %MathContainer117); Do(ff = %MathContainer120); Do(GG = %MathContainer118); Do(gg = %MathContainer121); Do(b = %MathContainer119); Do(a = %MathContainer122); if nops([FF])=0 then Do(%Plot17 = display(textplot([0,0,"Please provide F"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; if nops([ff])=0 then Do(%Plot17 = display(textplot([0,0,"Please provide f"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; if nops([GG])=0 then Do(%Plot17 = display(textplot([0,0,"Please provide G"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; if nops([gg])=0 then Do(%Plot17 = display(textplot([0,0,"Please provide g"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; if nops([b])= 0 then Do(%Plot17 = display(textplot([0,0,"Please provide b"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; if nops([a])= 0 then Do(%Plot17 = display(textplot([0,0,"Please provide a"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; p1 := display(textplot([0,0,"Check variables in f"],font=[TIMES,ROMAN,14]),axes=none); p2 := display(textplot([0,0,"Check variables in F"],font=[TIMES,ROMAN,14]),axes=none); p3 := display(textplot([0,0,"Check variables in g"],font=[TIMES,ROMAN,14]),axes=none); p4 := display(textplot([0,0,"Check variables in G"],font=[TIMES,ROMAN,14]),axes=none); p5 := display(textplot([0,0,"Lower limit 'a' must be numeric"],font=[TIMES,ROMAN,14]),axes=none); p6 := display(textplot([0,0,"Upper limit 'b' must be numeric"],font=[TIMES,ROMAN,14]),axes=none); if Do(%RadioButton12) then s2 := {r,theta}; s1 := {theta}; if nops(select(type,indets(ff),name) minus s2)>0 then Do(%MathContainer123 = p1);return;end if; if nops(select(type,indets(FF),name) minus s2)>0 then Do(%MathContainer123 = p2);return;end if; if nops(select(type,indets(gg),name) minus s1)>0 then Do(%MathContainer123 = p3);return;end if; if nops(select(type,indets(GG),name) minus s1)>0 then Do(%MathContainer123 = p4);return;end if; if not type(evalf(a),numeric) then Do(%MathContainer123 = p5);return;end if; if not type(evalf(b),numeric) then Do(%MathContainer123 = p6);return;end if; f := unapply(ff,(r,theta)); F := unapply(FF,(r,theta)); g := unapply(gg,theta); G := unapply(GG,theta); S1 := plot3d([r,theta,f(r,theta)],r=g(theta)..G(theta),theta=a..b,coords=cylindrical,color=blue): S2 := plot3d([r,theta,F(r,theta)],r=g(theta)..G(theta),theta=a..b,coords=cylindrical,color=red): S3 := plot3d([G(theta),theta,z],theta=a..b,z=f(G(theta),theta)..F(G(theta),theta),coords=cylindrical,color=gold): S4 := plot3d([g(theta),theta,z],theta=a..b,z=f(g(theta),theta)..F(g(theta),theta),coords=cylindrical,color=green): S5 := plot3d([r,a,z],r=g(a)..G(a),z=f(r,a)..F(r,a),coords=cylindrical,color=yellow): S6 := plot3d([r,b,z],r=g(b)..G(b),z=f(r,b)..F(r,b),coords=cylindrical,color=gray): elif Do(%RadioButton13) then s2 := {r,theta}; s1 := {r}; if nops(select(type,indets(ff),name) minus s2)>0 then Do(%MathContainer123 = p1);return;end if; if nops(select(type,indets(FF),name) minus s2)>0 then Do(%MathContainer123 = p2);return;end if; if nops(select(type,indets(gg),name) minus s1)>0 then Do(%MathContainer123 = p3);return;end if; if nops(select(type,indets(GG),name) minus s1)>0 then Do(%MathContainer123 = p4);return;end if; if not type(evalf(a),numeric) then Do(%MathContainer123 = p5);return;end if; if not type(evalf(b),numeric) then Do(%MathContainer123 = p6);return;end if; f := unapply(ff,(r,theta)); F := unapply(FF,(r,theta)); g := unapply(gg,r); G := unapply(GG,r); S1 := plot3d([r,theta,f(r,theta)],r=a..b,theta=g(r)..G(r),coords=cylindrical,color=blue): S2 := plot3d([r,theta,F(r,theta)],r=a..b,theta=g(r)..G(r),coords=cylindrical,color=red): S3 := plot3d([r,g(r),z],z=f(r,g(r))..F(r,g(r)),r=a..b,coords=cylindrical,color=green): S4 := plot3d([r,G(r),z],z=f(r,G(r))..F(r,G(r)),r=a..b,coords=cylindrical,color=gold): S5 := plot3d([a,theta,z],theta=g(a)..G(a),z=f(a,theta)..F(a,theta),coords=cylindrical,color=yellow): S6 := plot3d([b,theta,z],theta=g(b)..G(b),z=f(b,theta)..F(b,theta),coords=cylindrical,color=gray): elif Do(%RadioButton14) then s2 := {theta,z}; s1 := {z}; if nops(select(type,indets(ff),name) minus s2)>0 then Do(%MathContainer123 = p1);return;end if; if nops(select(type,indets(FF),name) minus s2)>0 then Do(%MathContainer123 = p2);return;end if; if nops(select(type,indets(gg),name) minus s1)>0 then Do(%MathContainer123 = p3);return;end if; if nops(select(type,indets(GG),name) minus s1)>0 then Do(%MathContainer123 = p4);return;end if; if not type(evalf(a),numeric) then Do(%MathContainer123 = p5);return;end if; if not type(evalf(b),numeric) then Do(%MathContainer123 = p6);return;end if; f := unapply(ff,(theta,z)); F := unapply(FF,(theta,z)); g := unapply(gg,z); G := unapply(GG,z); S1 := plot3d([f(theta,z),theta,z],theta=g(z)..G(z),z=a..b,coords=cylindrical,color=blue): S2 := plot3d([F(theta,z),theta,z],theta=g(z)..G(z),z=a..b,coords=cylindrical,color=red): S3 := plot3d([r,g(z),z],z=a..b,r=f(g(z),z)..F(g(z),z),coords=cylindrical,color=green): S4 := plot3d([r,G(z),z],z=a..b,r=f(G(z),z)..F(G(z),z),coords=cylindrical,color=gold): S5 := plot3d([r,theta,a],r=f(theta,a)..F(theta,a),theta=g(a)..G(a),coords=cylindrical,color=yellow): S6 := plot3d([r,theta,b],r=f(theta,b)..F(theta,b),theta=g(b)..G(b),coords=cylindrical,color=gray): elif Do(%RadioButton15) then s2 := {theta,z}; s1 := {theta}; if nops(select(type,indets(ff),name) minus s2)>0 then Do(%MathContainer123 = p1);return;end if; if nops(select(type,indets(FF),name) minus s2)>0 then Do(%MathContainer123 = p2);return;end if; if nops(select(type,indets(gg),name) minus s1)>0 then Do(%MathContainer123 = p3);return;end if; if nops(select(type,indets(GG),name) minus s1)>0 then Do(%MathContainer123 = p4);return;end if; if not type(evalf(a),numeric) then Do(%MathContainer123 = p5);return;end if; if not type(evalf(b),numeric) then Do(%MathContainer123 = p6);return;end if; f := unapply(ff,(theta,z)); F := unapply(FF,(theta,z)); g := unapply(gg,theta); G := unapply(GG,theta); S1 := plot3d([f(theta,z),theta,z],z=g(theta)..G(theta),theta=a..b,coords=cylindrical,color=blue): S2 := plot3d([F(theta,z),theta,z],z=g(theta)..G(theta),theta=a..b,coords=cylindrical,color=red): S3 := plot3d([r,theta,g(theta)],r=f(theta,g(theta))..F(theta,g(theta)),theta=a..b,coords=cylindrical,color=green): S4 := plot3d([r,theta,G(theta)],r=f(theta,G(theta))..F(theta,G(theta)),theta=a..b,coords=cylindrical,color=gold): S5 := plot3d([r,a,z],r=f(a,z)..F(a,z),z=g(a)..G(a),coords=cylindrical,color=yellow): S6 := plot3d([r,b,z],r=f(b,z)..F(b,z),z=g(b)..G(b),coords=cylindrical,color=gray): elif Do(%RadioButton16) then s2 := {r,z}; s1 := {z}; if nops(select(type,indets(ff),name) minus s2)>0 then Do(%MathContainer123 = p1);return;end if; if nops(select(type,indets(FF),name) minus s2)>0 then Do(%MathContainer123 = p2);return;end if; if nops(select(type,indets(gg),name) minus s1)>0 then Do(%MathContainer123 = p3);return;end if; if nops(select(type,indets(GG),name) minus s1)>0 then Do(%MathContainer123 = p4);return;end if; if not type(evalf(a),numeric) then Do(%MathContainer123 = p5);return;end if; if not type(evalf(b),numeric) then Do(%MathContainer123 = p6);return;end if; f := unapply(ff,(r,z)); F := unapply(FF,(r,z)); g := unapply(gg,z); G := unapply(GG,z); S1 := plot3d([r,f(r,z),z],r=g(z)..G(z),z=a..b,coords=cylindrical,color=blue): S2 := plot3d([r,F(r,z),z],r=g(z)..G(z),z=a..b,coords=cylindrical,color=red): S3 := plot3d([g(z),theta,z],theta=f(g(z),z)..F(g(z),z),z=a..b,coords=cylindrical,color=green): S4 := plot3d([G(z),theta,z],theta=f(G(z),z)..F(G(z),z),z=a..b,coords=cylindrical,color=gold): S5 := plot3d([r,theta,a],theta=f(r,a)..F(r,a),r=g(a)..G(a),coords=cylindrical,color=yellow): S6 := plot3d([r,theta,b],theta=f(r,b)..F(r,b),r=g(b)..G(b),coords=cylindrical,color=gray): elif Do(%RadioButton17) then s2 := {r,z}; s1 := {r}; if nops(select(type,indets(ff),name) minus s2)>0 then Do(%MathContainer123 = p1);return;end if; if nops(select(type,indets(FF),name) minus s2)>0 then Do(%MathContainer123 = p2);return;end if; if nops(select(type,indets(gg),name) minus s1)>0 then Do(%MathContainer123 = p3);return;end if; if nops(select(type,indets(GG),name) minus s1)>0 then Do(%MathContainer123 = p4);return;end if; if not type(evalf(a),numeric) then Do(%MathContainer123 = p5);return;end if; if not type(evalf(b),numeric) then Do(%MathContainer123 = p6);return;end if; f := unapply(ff,(r,z)); F := unapply(FF,(r,z)); g := unapply(gg,r); G := unapply(GG,r); S1 := plot3d([r,f(r,z),z],r=a..b,z=g(r)..G(r),coords=cylindrical,color=blue): S2 := plot3d([r,F(r,z),z],r=a..b,z=g(r)..G(r),coords=cylindrical,color=red): S3 := plot3d([r,theta,g(r)],theta=f(r,g(r))..F(r,g(r)),r=a..b,coords=cylindrical,color=green): S4 := plot3d([r,theta,G(r)],theta=f(r,G(r))..F(r,G(r)),r=a..b,coords=cylindrical,color=green): S5 := plot3d([a,theta,z],theta=f(a,z)..F(a,z),z=g(a)..G(a),coords=cylindrical,color=yellow): S6 := plot3d([b,theta,z],theta=f(b,z)..F(b,z),z=g(b)..G(b),coords=cylindrical,color=gray): else return; end if; P := display([S1,S2,S3,S4,S5,S6],axes=box,labels=[x,y,z],orientation=[-115,80]); Do(%Plot17 = P); end module; end use;

use DocumentTools, plots in Do(%Plot17 = display(textplot([0,0," "]),axes=none)); end use;

use DocumentTools, plots in Do(%MathContainer115 = NULL); Do(%MathContainer116 = NULL); Do(%MathContainer117 = NULL); Do(%MathContainer120 = NULL); Do(%MathContainer118 = NULL); Do(%MathContainer121 = NULL); Do(%MathContainer119 = NULL); Do(%MathContainer122 = NULL); Do(%MathContainer123 = NULL); Do(%Plot17 = display(textplot([0,0," "]),axes=none)); Do(%RadioButton11(value)=true); end use;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Tabela 32 Modelo de tarefas para uma integral tripla iterada em coordenadas cil\303\255ndricas

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Ao passo que a integra\303\247\303\243o na dire\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= precede a integra\303\247\303\243o na dire\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, o volume pode ser calculado com uma \303\272nica integral tripla.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 3" layout="Heading 3">Coordenadas Cartesianas</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Use coordenadas cartesianas para calcular o volume do s\303\263lido tirado de uma esfera unit\303\241ria cortada por um cone cujo v\303\251rtice est\303\241 no centro da esfera e cujo gerador faz um \303\242ngulo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEjMzBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21vR0YkNi1RKSZjb21wZm47RidGMi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjE2NjY2NjdlbUYnLyUncnNwYWNlR0ZMLyUnaXRhbGljR1EldHJ1ZUYnL0YzUSdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGMg==com o eixo de simetria do cone.

O modelo de tarefa na Tabela 33 providencia uma solu\303\247\303\243o. A localiza\303\247\303\243o no modelo de tarefa \303\251 dado no topo da tabela.

Ferramentas\342\211\273Tarefas\342\211\273Pesquisar:

Calculus - Multivariate\342\211\273Integration\342\211\273Visualizing Regions of Integration\342\211\273Spherical

Calcule 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 e fa\303\247a o Gr\303\241fico de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=

Elemento de Volume 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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYlUScmIzk2MTtGJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRictSSNtb0dGJDYtUSJ+RidGOi8lJmZlbmNlR0Y5LyUqc2VwYXJhdG9yR0Y5LyUpc3RyZXRjaHlHRjkvJSpzeW1tZXRyaWNHRjkvJShsYXJnZW9wR0Y5LyUubW92YWJsZWxpbWl0c0dGOS8lJ2FjY2VudEdGOS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlBGKy1GLDYlUScmIzk2NjtGJ0Y4RjpGPEYrLUYsNiVRJyYjOTUyO0YnRjhGOkY6

use DocumentTools in module() local rho,phi,theta,Psi,g,G,f,F,a,b; Do(%MathContainer124 = Int(Psi*rho^2*sin(phi),[rho=f(phi,theta)..F(phi,theta), phi=g(theta)..G(theta), theta=a..b])); end module; end use;

use DocumentTools in module() local rho,phi,theta,Psi,g,G,f,F,a,b; Do(%MathContainer124 = Int(Psi*rho^2*sin(phi),[rho=f(phi,theta)..F(phi,theta), theta=g(phi)..G(phi), phi=a..b])); end module; end use;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYlUScmIzk2NjtGJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRictSSNtb0dGJDYtUSJ+RidGOi8lJmZlbmNlR0Y5LyUqc2VwYXJhdG9yR0Y5LyUpc3RyZXRjaHlHRjkvJSpzeW1tZXRyaWNHRjkvJShsYXJnZW9wR0Y5LyUubW92YWJsZWxpbWl0c0dGOS8lJ2FjY2VudEdGOS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlBGKy1GLDYlUScmIzk2MTtGJ0Y4RjpGPEYrLUYsNiVRJyYjOTUyO0YnRjhGOkY6

use DocumentTools in module() local rho,phi,theta,Psi,g,G,f,F,a,b; Do(%MathContainer124 = Int(Psi*rho^2*sin(phi),[phi=f(rho,theta)..F(rho,theta), rho=g(theta)..G(theta), theta=a..b])); end module; end use;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYlUScmIzk2NjtGJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRictSSNtb0dGJDYtUSJ+RidGOi8lJmZlbmNlR0Y5LyUqc2VwYXJhdG9yR0Y5LyUpc3RyZXRjaHlHRjkvJSpzeW1tZXRyaWNHRjkvJShsYXJnZW9wR0Y5LyUubW92YWJsZWxpbWl0c0dGOS8lJ2FjY2VudEdGOS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlBGKy1GLDYlUScmIzk1MjtGJ0Y4RjpGPEYrLUYsNiVRJyYjOTYxO0YnRjhGOkY6

use DocumentTools in module() local rho,phi,theta,Psi,g,G,f,F,a,b; Do(%MathContainer124 = Int(Psi*rho^2*sin(phi),[phi=f(rho,theta)..F(rho,theta), theta=g(rho)..G(rho), rho=a..b])); end module end use;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYlUScmIzk1MjtGJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRictSSNtb0dGJDYtUSJ+RidGOi8lJmZlbmNlR0Y5LyUqc2VwYXJhdG9yR0Y5LyUpc3RyZXRjaHlHRjkvJSpzeW1tZXRyaWNHRjkvJShsYXJnZW9wR0Y5LyUubW92YWJsZWxpbWl0c0dGOS8lJ2FjY2VudEdGOS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlBGKy1GLDYlUScmIzk2NjtGJ0Y4RjpGPEYrLUYsNiVRJyYjOTYxO0YnRjhGOkY6

use DocumentTools in module() local rho,phi,theta,Psi,g,G,f,F,a,b; Do(%MathContainer124 = Int(Psi*rho^2*sin(phi),[theta=f(rho,phi)..F(rho,phi), phi=g(rho)..G(rho), rho=a..b])); end module; end use;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYlUScmIzk1MjtGJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRictSSNtb0dGJDYtUSJ+RidGOi8lJmZlbmNlR0Y5LyUqc2VwYXJhdG9yR0Y5LyUpc3RyZXRjaHlHRjkvJSpzeW1tZXRyaWNHRjkvJShsYXJnZW9wR0Y5LyUubW92YWJsZWxpbWl0c0dGOS8lJ2FjY2VudEdGOS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlBGKy1GLDYlUScmIzk2MTtGJ0Y4RjpGPEYrLUYsNiVRJyYjOTY2O0YnRjhGOkY6

use DocumentTools in module() local rho,phi,theta,Psi,g,G,f,F,a,b; Do(%MathContainer124 = Int(Psi*rho^2*sin(phi),[theta=f(rho,phi)..F(rho,phi), rho=g(phi)..G(phi), phi=a..b])); end module; end use;

use DocumentTools in Do(%MathContainer125 = NULL); end use;

use DocumentTools in Do(%MathContainer126 = NULL); end use;

use DocumentTools in Do(%MathContainer127 = NULL); end use;

use DocumentTools in Do(%MathContainer128 = NULL); end use;

use DocumentTools in Do(%MathContainer129 = NULL); end use;

use DocumentTools in Do(%MathContainer130 = NULL); end use;

use DocumentTools in Do(%MathContainer131 = NULL); end use;

use DocumentTools in module() local f,F,g,G,a,b,q, P, s1,s2; if Do(%RadioButton18) then Do(%MathContainer124 = MathML[Export]("Select a volume element"));return;end if; Do(F = %MathContainer126); Do(f = %MathContainer129); Do(G = %MathContainer127); Do(g = %MathContainer130); Do(b = %MathContainer128); Do(a = %MathContainer131); Do(P = %MathContainer125); if nops([P])=0 then Do(%MathContainer132 = MathML[Export]("Please provide an integrand")); return; end if; if nops([F])=0 then Do(%MathContainer132 = MathML[Export]("Please provide F")); return; end if; if nops([f])=0 then Do(%MathContainer132 = MathML[Export]("Please provide f")); return; end if; if nops([G])=0 then Do(%MathContainer132 = MathML[Export]("Please provide G")); return; end if; if nops([g])=0 then Do(%MathContainer132 = MathML[Export]("Please provide g")); return; end if; if nops([b])=0 then Do(%MathContainer132 = MathML[Export]("Please provide b")); return; end if; if nops([a])=0 then Do(%MathContainer132 = MathML[Export]("Please provide a")); return; end if; if Do(%RadioButton19) then s2 := {phi,theta}; s1 := {theta}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer132 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer132 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer132 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer132 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer132 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer132 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P*rho^2*sin(phi),[rho=f..F, phi=g..G, theta=a..b]); elif Do(%RadioButton20) then s2 := {phi,theta}; s1 := {phi}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer132 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer132 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer132 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer132 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer132 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer132 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P*rho^2*sin(phi),[rho=f..F, theta=g..G, phi=a..b]); elif Do(%RadioButton21) then s2 := {rho,theta}; s1 := {theta}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer132 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer132 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer132 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer132 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer132 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer132 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P*rho^2*sin(phi),[phi=f..F, rho=g..G, theta=a..b]); elif Do(%RadioButton22) then s2 := {rho,theta}; s1 := {rho}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer132 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer132 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer132 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer132 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer132 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer132 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P*rho^2*sin(phi),[phi=f..F, theta=g..G, rho=a..b]); elif Do(%RadioButton23) then s2 := {rho,phi}; s1 := {phi}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer132 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer132 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer132 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer132 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer132 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer132 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P*rho^2*sin(phi),[theta=f..F, phi=g..G, rho=a..b]); elif Do(%RadioButton24) then s2 := {rho,phi}; s1 := {phi}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer132 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer132 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer132 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer132 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer132 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer132 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P*rho^2*sin(phi),[theta=f..F, rho=g..G, phi=a..b]); else return; end if; Do(%MathContainer132 = value(q)); end module; end use;

use DocumentTools in module() local f,F,g,G,a,b,q, P; if Do(%RadioButton18) then Do(%MathContainer124 = MathML[Export]("Select a volume element"));return;end if; Do(F = %MathContainer126); Do(f = %MathContainer129); Do(G = %MathContainer127); Do(g = %MathContainer130); Do(b = %MathContainer128); Do(a = %MathContainer131); Do(P = %MathContainer125); if nops([F])=0 then Do(%MathContainer132 = MathML[Export]("Please provide F")); return; end if; if nops([f])=0 then Do(%MathContainer132 = MathML[Export]("Please provide f")); return; end if; if nops([G])=0 then Do(%MathContainer132 = MathML[Export]("Please provide G")); return; end if; if nops([g])=0 then Do(%MathContainer132 = MathML[Export]("Please provide g")); return; end if; if nops([b])=0 then Do(%MathContainer132 = MathML[Export]("Please provide b")); return; end if; if nops([a])=0 then Do(%MathContainer132 = MathML[Export]("Please provide a")); return; end if; if nops([P])=0 then Do(%MathContainer132 = MathML[Export]("Please provide an integrand")); return; end if; if Do(%RadioButton19) then s2 := {phi,theta}; s1 := {theta}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer132 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer132 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer132 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer132 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer132 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer132 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P*rho^2*sin(phi),[rho=f..F, phi=g..G, theta=a..b]); elif Do(%RadioButton20) then s2 := {phi,theta}; s1 := {phi}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer132 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer132 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer132 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer132 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer132 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer132 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P*rho^2*sin(phi),[rho=f..F, theta=g..G, phi=a..b]); elif Do(%RadioButton21) then s2 := {rho,theta}; s1 := {theta}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer132 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer132 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer132 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer132 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer132 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer132 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P*rho^2*sin(phi),[phi=f..F, rho=g..G, theta=a..b]); elif Do(%RadioButton22) then s2 := {rho,theta}; s1 := {rho}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer132 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer132 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer132 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer132 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer132 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer132 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P*rho^2*sin(phi),[phi=f..F, theta=g..G, rho=a..b]); elif Do(%RadioButton23) then s2 := {rho,phi}; s1 := {rho}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer132 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer132 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer132 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer132 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer132 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer132 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P*rho^2*sin(phi),[theta=f..F, phi=g..G, rho=a..b]); elif Do(%RadioButton24) then s2 := {rho,phi}; s1 := {phi}; if nops(select(type,indets(f),name) minus s2)>0 then Do(%MathContainer132 = MathML[Export]("Check variables in f"));return;end if; if nops(select(type,indets(F),name) minus s2)>0 then Do(%MathContainer132 = MathML[Export]("Check variables in F"));return;end if; if nops(select(type,indets(g),name) minus s1)>0 then Do(%MathContainer132 = MathML[Export]("Check variable in g"));return;end if; if nops(select(type,indets(G),name) minus s1)>0 then Do(%MathContainer132 = MathML[Export]("Check variable in G"));return;end if; if not type(evalf(a),numeric) then Do(%MathContainer132 = MathML[Export]("Lower limit 'a' must be numeric"));return;end if; if not type(evalf(b),numeric) then Do(%MathContainer132 = MathML[Export]("Upper limit 'b' must be numeric"));return;end if; q := Int(P*rho^2*sin(phi),[theta=f..F, rho=g..G, phi=a..b]); else return; end if; Do(%MathContainer132 = evalf(value(q))); end module; end use;

use DocumentTools, plots in module() local f,F,g,G,a,b, ff, FF, gg, GG, P, S1,S2,S3,S4,S5,S6, s2, s1, p1,p2,p3,p4,p5,p6; if Do(%RadioButton18) then Do(%Plot18 = display(textplot([0,0,"Please select a volume element"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; Do(FF = %MathContainer126); Do(ff = %MathContainer129); Do(GG = %MathContainer127); Do(gg = %MathContainer130); Do(b = %MathContainer128); Do(a = %MathContainer131); if nops([FF])=0 then Do(%Plot18 = display(textplot([0,0,"Please provide F"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; if nops([ff])=0 then Do(%Plot18 = display(textplot([0,0,"Please provide f"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; if nops([GG])=0 then Do(%Plot18 = display(textplot([0,0,"Please provide G"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; if nops([gg])=0 then Do(%Plot18 = display(textplot([0,0,"Please provide g"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; if nops([b])= 0 then Do(%Plot18 = display(textplot([0,0,"Please provide b"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; if nops([a])= 0 then Do(%Plot18 = display(textplot([0,0,"Please provide a"],font=[TIMES,ROMAN,14]),axes=none)); return; end if; p1 := display(textplot([0,0,"Check variables in f"],font=[TIMES,ROMAN,14]),axes=none); p2 := display(textplot([0,0,"Check variables in F"],font=[TIMES,ROMAN,14]),axes=none); p3 := display(textplot([0,0,"Check variables in g"],font=[TIMES,ROMAN,14]),axes=none); p4 := display(textplot([0,0,"Check variables in G"],font=[TIMES,ROMAN,14]),axes=none); p5 := display(textplot([0,0,"Lower limit 'a' must be numeric"],font=[TIMES,ROMAN,14]),axes=none); p6 := display(textplot([0,0,"Upper limit 'b' must be numeric"],font=[TIMES,ROMAN,14]),axes=none); if Do(%RadioButton19) then s2 := {phi,theta}; s1 := {theta}; if nops(select(type,indets(ff),name) minus s2)>0 then Do(%MathContainer132 = p1);return;end if; if nops(select(type,indets(FF),name) minus s2)>0 then Do(%MathContainer132 = p2);return;end if; if nops(select(type,indets(gg),name) minus s1)>0 then Do(%MathContainer132 = p3);return;end if; if nops(select(type,indets(GG),name) minus s1)>0 then Do(%MathContainer132 = p4);return;end if; if not type(evalf(a),numeric) then Do(%MathContainer132 = p5);return;end if; if not type(evalf(b),numeric) then Do(%MathContainer132 = p6);return;end if; f := unapply(ff,(phi,theta)); F := unapply(FF,(phi,theta)); g := unapply(gg,theta); G := unapply(GG,theta); S1 := plot3d([f(p,t),t,p],p=g(t)..G(t),t=a..b,coords=spherical,color=blue): S2 := plot3d([F(p,t),t,p],p=g(t)..G(t),t=a..b,coords=spherical,color=red): S3 := plot3d([r,t,g(t)],t=a..b,r=f(g(t),t)..F(g(t),t),coords=spherical,color=green): S4 := plot3d([r,t,G(t)],t=a..b,r=f(G(t),t)..F(G(t),t),coords=spherical,color=gold): S5 := plot3d([r,a,p],p=g(a)..G(a),r=f(p,a)..F(p,a),coords=spherical,color=yellow): S6 := plot3d([r,b,p],p=g(b)..G(b),r=f(p,b)..F(p,b),coords=spherical,color=gray): elif Do(%RadioButton20) then s2 := {phi,theta}; s1 := {phi}; if nops(select(type,indets(ff),name) minus s2)>0 then Do(%MathContainer132 = p1);return;end if; if nops(select(type,indets(FF),name) minus s2)>0 then Do(%MathContainer132 = p2);return;end if; if nops(select(type,indets(gg),name) minus s1)>0 then Do(%MathContainer132 = p3);return;end if; if nops(select(type,indets(GG),name) minus s1)>0 then Do(%MathContainer132 = p4);return;end if; if not type(evalf(a),numeric) then Do(%MathContainer132 = p5);return;end if; if not type(evalf(b),numeric) then Do(%MathContainer132 = p6);return;end if; f := unapply(ff,(phi,theta)); F := unapply(FF,(phi,theta)); g := unapply(gg,phi); G := unapply(GG,phi); S1 := plot3d([f(p,t),t,p],t=g(p)..G(p),p=a..b,coords=spherical,color=blue): S2 := plot3d([F(p,t),t,p],t=g(p)..G(p),p=a..b,coords=spherical,color=red): S3 := plot3d([r,g(p),p],r=f(p,g(p))..F(p,g(p)),p=a..b,coords=spherical,color=green): S4 := plot3d([r,G(p),p],r=f(p,G(p))..F(p,G(p)),p=a..b,coords=spherical,color=gold): S5 := plot3d([r,t,a],r=f(a,t)..F(a,t),t=g(a)..G(a),coords=spherical,color=yellow): S6 := plot3d([r,t,b],r=f(b,t)..F(b,t),t=g(b)..G(b),coords=spherical,color=gray): elif Do(%RadioButton21) then s2 := {rho,theta}; s1 := {theta}; if nops(select(type,indets(ff),name) minus s2)>0 then Do(%MathContainer132 = p1);return;end if; if nops(select(type,indets(FF),name) minus s2)>0 then Do(%MathContainer132 = p2);return;end if; if nops(select(type,indets(gg),name) minus s1)>0 then Do(%MathContainer132 = p3);return;end if; if nops(select(type,indets(GG),name) minus s1)>0 then Do(%MathContainer132 = p4);return;end if; if not type(evalf(a),numeric) then Do(%MathContainer132 = p5);return;end if; if not type(evalf(b),numeric) then Do(%MathContainer132 = p6);return;end if; f := unapply(ff,(rho,theta)); F := unapply(FF,(rho,theta)); g := unapply(gg,theta); G := unapply(GG,theta); S1 := plot3d([r,t,f(r,t)],r=g(t)..G(t),t=a..b,coords=spherical,color=blue): S2 := plot3d([r,t,F(r,t)],r=g(t)..G(t),t=a..b,coords=spherical,color=red): S3 := plot3d([g(t),t,p],p=f(g(t),t)..F(g(t),t),t=a..b,coords=spherical,color=green): S4 := plot3d([G(t),t,p],p=f(G(t),t)..F(G(t),t),t=a..b,coords=spherical,color=gold): S5 := plot3d([r,a,p],r=g(a)..G(a),p=f(r,a)..F(r,a),coords=spherical,color=yellow): S6 := plot3d([r,b,p],r=g(b)..G(b),p=f(r,b)..F(r,b),coords=spherical,color=gray): elif Do(%RadioButton22) then s2 := {rho,theta}; s1 := {rho}; if nops(select(type,indets(ff),name) minus s2)>0 then Do(%MathContainer132 = p1);return;end if; if nops(select(type,indets(FF),name) minus s2)>0 then Do(%MathContainer132 = p2);return;end if; if nops(select(type,indets(gg),name) minus s1)>0 then Do(%MathContainer132 = p3);return;end if; if nops(select(type,indets(GG),name) minus s1)>0 then Do(%MathContainer132 = p4);return;end if; if not type(evalf(a),numeric) then Do(%MathContainer132 = p5);return;end if; if not type(evalf(b),numeric) then Do(%MathContainer132 = p6);return;end if; f := unapply(ff,(rho,theta)); F := unapply(FF,(rho,theta)); g := unapply(gg,rho); G := unapply(GG,rho); S1 := plot3d([r,t,f(r,t)],t=g(r)..G(r),r=a..b,coords=spherical,color=blue): S2 := plot3d([r,t,F(r,t)],t=g(r)..G(r),r=a..b,coords=spherical,color=blue): S3 := plot3d([r,g(r),p],p=f(r,g(r))..F(r,g(r)),r=a..b,coords=spherical,color=green): S4 := plot3d([r,G(r),p],p=f(r,G(r))..F(r,G(r)),r=a..b,coords=spherical,color=gold): S5 := plot3d([a,t,p],t=g(a)..G(a),p=f(a,t)..F(a,t),coords=spherical,color=yellow): S6 := plot3d([b,t,p],t=g(b)..G(b),p=f(b,t)..F(b,t),coords=spherical,color=gray): elif Do(%RadioButton23) then s2 := {rho,phi}; s1 := {rho}; if nops(select(type,indets(ff),name) minus s2)>0 then Do(%MathContainer132 = p1);return;end if; if nops(select(type,indets(FF),name) minus s2)>0 then Do(%MathContainer132 = p2);return;end if; if nops(select(type,indets(gg),name) minus s1)>0 then Do(%MathContainer132 = p3);return;end if; if nops(select(type,indets(GG),name) minus s1)>0 then Do(%MathContainer132 = p4);return;end if; if not type(evalf(a),numeric) then Do(%MathContainer132 = p5);return;end if; if not type(evalf(b),numeric) then Do(%MathContainer132 = p6);return;end if; f := unapply(ff,(rho,phi)); F := unapply(FF,(rho,phi)); g := unapply(gg,rho); G := unapply(GG,rho); S1 := plot3d([r,f(r,p),p],p=g(r)..G(r),r=a..b,coords=spherical,color=blue): S2 := plot3d([r,F(r,p),p],p=g(r)..G(r),r=a..b,coords=spherical,color=red): S3 := plot3d([r,t,g(r)],r=a..b,t=f(r,g(r))..F(r,g(r)),coords=spherical,color=green): S4 := plot3d([r,t,G(r)],r=a..b,t=f(r,G(r))..F(r,G(r)),coords=spherical,color=gold): S5 := plot3d([a,t,p],p=g(a)..G(a),t=f(a,p)..F(a,p),coords=spherical,color=yellow): S6 := plot3d([b,t,p],p=g(b)..G(b),t=f(b,p)..F(b,p),coords=spherical,color=gray): elif Do(%RadioButton24) then s2 := {rho,phi}; s1 := {phi}; if nops(select(type,indets(ff),name) minus s2)>0 then Do(%MathContainer132 = p1);return;end if; if nops(select(type,indets(FF),name) minus s2)>0 then Do(%MathContainer132 = p2);return;end if; if nops(select(type,indets(gg),name) minus s1)>0 then Do(%MathContainer132 = p3);return;end if; if nops(select(type,indets(GG),name) minus s1)>0 then Do(%MathContainer132 = p4);return;end if; if not type(evalf(a),numeric) then Do(%MathContainer132 = p5);return;end if; if not type(evalf(b),numeric) then Do(%MathContainer132 = p6);return;end if; f := unapply(ff,(rho,phi)); F := unapply(FF,(rho,phi)); g := unapply(gg,phi); G := unapply(GG,phi); S1 := plot3d([r,f(r,p),p],r=g(p)..G(p),p=a..b,coords=spherical,color=blue): S2 := plot3d([r,F(r,p),p],r=g(p)..G(p),p=a..b,coords=spherical,color=red): S3 := plot3d([g(p),t,p],t=f(g(p),p)..F(g(p),p),p=a..b,coords=spherical,color=green): S4 := plot3d([G(p),t,p],t=f(G(p),p)..F(G(p),p),p=a..b,coords=spherical,color=gold): S5 := plot3d([r,t,a],r=g(a)..G(a),t=f(r,a)..F(r,a),coords=spherical,color=yellow): S6 := plot3d([r,t,b],r=g(b)..G(b),t=f(r,b)..F(r,b),coords=spherical,color=yellow): else return; end if; P := display([S1,S2,S3,S4,S5,S6],axes=box,labels=[x,y,z],orientation=[-115,80]); Do(%Plot18 = P); end module; end use;

use DocumentTools, plots in Do(%Plot18 = display(textplot([0,0," "]),axes=none)); end use;

use DocumentTools, plots in Do(%MathContainer124 = NULL); Do(%MathContainer125 = NULL); Do(%MathContainer126 = NULL); Do(%MathContainer129 = NULL); Do(%MathContainer127 = NULL); Do(%MathContainer130 = NULL); Do(%MathContainer128 = NULL); Do(%MathContainer131 = NULL); Do(%MathContainer132 = NULL); Do(%Plot18 = display(textplot([0,0," "]),axes=none)); Do(%RadioButton18(value)=true); end use;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Tabela 33 Modelo de tarefas para uma integral tripla iterada em coordenadas esf\303\251ricas

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 2" layout="Heading 2">Pontos Cr\303\255ticos e o Teste da Segunda Derivada</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Uma tarefa comum em um curso de c\303\241lculo de m\303\272ltiplas vari\303\241veis \303\251 a determina\303\247\303\243o e classifica\303\247\303\243o dos pontos cr\303\255ticos de uma fun\303\247\303\243o multivariada. A tabela 34 mostra isso com um modelo de tarefa.

Ferramentas\342\211\273Tarefas\342\211\273Pesquisar: Multivariate Calculus\342\211\273Critical Points & Second Derivative Test

Pontos Cr\303\255ticos e o Teste da Segunda Derivada

Fun\303\247\303\243o Objetiva LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=

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

LCgqJClJInhHNiIiIiMiIiJGKCooIiIkRihGJUYoKUkieUdGJkYnRihGKComIiImRigpRixGKkYoISIi

Lista de Vari\303\241veis Independentes

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NyRJInhHNiJJInlHRiQ=

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Pontos Cr\303\255ticos

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Tste da Segunda Derivada

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Hessianas e seus Autovalores

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Tabela 34 Descobrindo e classificando pontos cr\303\255ticos de uma fun\303\247\303\243o multivariada

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A fun\303\247\303\243o dada possui dois pontos cr\303\255ticos, ambos descobertos com o comando Solve. Entretanto, o formato da solu\303\247\303\243o n\303\243o s\303\243o "pontos" tal que a sa\303\255da deve ser colocada na forma de uma lista de listas. O teste da segunda derivada \303\251 aplicado em cada ponto. A origem n\303\243o pode ser classificada por esse teste, ent\303\243o nada pode ser dito sobre este teste. O outro ponto ponto \303\251 um ponto de sela. Na "linha" final do modelo, a matriz Hessiana (a matriz das segindas derivadas) e seus autovalores s\303\243o dados para cada ponto. Como a Hessiana \303\251 sim\303\251trica, os sinais de seus autovalores s\303\243o suficientes para determinar se a matriz \303\251 positiva definida ou negativa definida ou mesmo indefinida. Na origem, a Hessiana possui um autovalor zero e \303\251 singular. Isto \303\251 o porqu\303\252 da origem n\303\243o poder ser classificada pelo teste da segunda derivada . Os autovalores no outro ponto s\303\243o de sinais opostos; ent\303\243o a Hessiana \303\251 indefinida. Por isso o segundo ponto \303\251 um ponto de sela.

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<Text-field style="Heading 2" layout="Heading 2">Centro de Massa</Text-field>

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O pacote Student Precalculus cont\303\251m o comando CenterOfMass que vai determinar o centro de massa de uma dstribui\303\247\303\243o discreta de massas em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJuRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1. O pacote Student Multivariate Calculus cont\303\251m um comando CenterOfMass que ir\303\241 determinar o centro de massa de uma distribui\303\247\303\243o cont\303\255nua de massa em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 ou LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjNGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1, usando coordenadas Cartesianas, polares, esf\303\251ricas, ou cil\303\255ndricas. Em cada caso, este comando escreve as express\303\265es para as coordenadas do centro de massa, ent\303\243o calcula a integral expressando os momentos apropriados e massa total. Em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 (Cartesiano e polar), o comando CenterOfMass pode desenhar um gr\303\241fico de uma fun\303\247\303\243o densidade sobre uma regi\303\243o plana na qual \303\251 definida. Todos os casos cont\303\255nuos s\303\243o implementados em modelos de tarefas.

<Text-field style="Heading 3" layout="Heading 3">Coordenadas Cartesianas 2-D </Text-field>

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Para descobrir o centro de massa da regi\303\243o plana

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cuja densidade \303\251 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, use o modelode tarefa na Tabela 35.

Ferramentas\342\211\273Tarefas\342\211\273Pesquisar: Multivariate Calculus\342\211\273Center of Mass\342\211\273Cartesian 2-D

Centro de Massa para uma Regi\303\243o Plana em Coordenadas Cartesianas

Densidade:

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KiYpSSJ4RzYiIiIjIiIiSSJ5R0YlRic=

KiQpSSJ4RzYiIiIjIiIi

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KiQpSSJ4RzYiIyIiIiIiI0Yn

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IiIh

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IiIi

MomentosLUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JC1JI21vR0YkNi1RJiZkaXY7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y7LyUpc3RyZXRjaHlHRjsvJSpzeW1tZXRyaWNHRjsvJShsYXJnZW9wR0Y7LyUubW92YWJsZWxpbWl0c0dGOy8lJ2FjY2VudEdGOy8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkpGNkYrRjY=Massa:

Integral Inerte - LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2Ji1GLDYlUSNkeUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GOVEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkMvJSlzdHJldGNoeUdGQy8lKnN5bW1ldHJpY0dGQy8lKGxhcmdlb3BHRkMvJS5tb3ZhYmxlbGltaXRzR0ZDLyUnYWNjZW50R0ZDLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUi1GLDYlUSNkeEYnRjVGOEY/RitGPw==

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Valores Explicitos para LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JC1JJm1vdmVyR0YkNiYtRjA2JC1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y+USdub3JtYWxGJy1JI21vR0YkNi1RLSZjb25qdWdhdGUwO0YnRkAvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkgvJSlzdHJldGNoeUdGPC8lKnN5bW1ldHJpY0dGSC8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZILyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGV0ZTL0krbXNlbWFudGljc0dGJFEqY29uanVnYXRlRidGQEYrRkA= e LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JC1JJm1vdmVyR0YkNiYtRjA2JC1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y+USdub3JtYWxGJy1JI21vR0YkNi1RLSZjb25qdWdhdGUwO0YnRkAvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkgvJSlzdHJldGNoeUdGPC8lKnN5bW1ldHJpY0dGSC8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZILyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGV0ZTL0krbXNlbWFudGljc0dGJFEqY29uanVnYXRlRidGQEYrRkA=

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Gr\303\241fico:

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LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNictSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JUYrLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRitGMkY1

Tabela 35 Centro de massa de uma regi\303\243o plana em coordenadas Cartesianas

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

A regi\303\243o em vermelho no gr\303\241fico \303\251 a regi\303\243o plana cujo centro de massa est\303\241 localizado no ponto de cor verde, e a superf\303\255cie azul \303\251 o gr\303\241fico da fun\303\247\303\243o densidade LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieUYnRj1GP0YyRjJGMg==.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 3" layout="Heading 3">Polar</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Para encontrar o centro de massa de uma regi\303\243o plana

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

cuja densidade \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJW1zdXBHRiQ2JS1GLDYlUSJyRicvRjBRJXRydWVGJy9GM1EnaXRhbGljRictRiM2JS1JI21uR0YkNiRRIjJGJ0YyRlJGVC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGMg==, use o modelo de tarefa na Tabela36.

Ferramentas\342\211\273Tarefas\342\211\273Pesquisar: Multivariate Calculus\342\211\273Center of Mass\342\211\273Polar

Centro de Massa para uma Regi\303\243o Plana em Coordenadas Polares

Densidade:

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KiQpSSJyRzYiIiIjIiIi

Regi\303\243o: 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IiIi

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IiIj

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Integral Inerte- LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2Jy1GLDYlUSNkckYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GOVEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkMvJSlzdHJldGNoeUdGQy8lKnN5bW1ldHJpY0dGQy8lKGxhcmdlb3BHRkMvJS5tb3ZhYmxlbGltaXRzR0ZDLyUnYWNjZW50R0ZDLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUi1GLDYlUSJkRidGNUY4LUYsNiVRJyYjOTUyO0YnL0Y2RkNGP0Y/RitGPw==

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Gr\303\241fico:

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Tabela 36 Centro de massa de uma regi\303\243o plana em coordenadas polares

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A regi\303\243o em vermelho no gr\303\241fico \303\251 a regi\303\243o plana cujo centro de massa est\303\241 localizado no ponto de cor verde, e a superf\303\255cie de cor azul \303\251 o gr\303\241fico da fun\303\247\303\243o densidade LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInJGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEnJiM5NTI7RidGL0YyRjJGMkYy.

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<Text-field style="Heading 3" layout="Heading 3">Coordenadas Cartesianas 3-D</Text-field>

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Parad encontrar o centro de massa da regi\303\243o

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cuja densidade \303\251 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, use o modelo de tarefa na Tabela 37.

Ferramentas\342\211\273Tarefas\342\211\273Pesquisar: Multivariate Calculus\342\211\273Center of Mass\342\211\273Cartesian 3-D

Centro de Massa de uma Regi\303\243o 3D em Coordenadas Cartesianas

Densidade:

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Regi\303\243o: LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JC1JKG1mZW5jZWRHRiQ2Ji1GMDY4LUklbXN1YkdGJDYlLUYsNiVRInpGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRjA2JC1JI21uR0YkNiRRIjFGJy9GQVEnbm9ybWFsRidGSS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUYzNiQtRjA2Ji1GLDYlUSJ4RidGPUZALUkjbW9HRiQ2LVEiLEYnRkkvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj8vJSlzdHJldGNoeUdGZW4vJSpzeW1tZXRyaWNHRmVuLyUobGFyZ2VvcEdGZW4vJS5tb3ZhYmxlbGltaXRzR0Zlbi8lJ2FjY2VudEdGZW4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSJ5RidGPUZARklGSS1GVjYtUSYmbGVxO0YnRklGWS9GZ25GZW5GaG5Gam5GXG9GXm9GYG8vRmNvUSwwLjI3Nzc3NzhlbUYnL0Zmb0ZgcEY6RltwLUY4NiVGOi1GMDYkLUZGNiRRIjJGJ0ZJRklGS0ZORlUtRjg2JUZob0ZDRkstRjM2JC1GMDYkRlJGSUZJRltwRmhvRltwLUY4NiVGaG9GZHBGS0ZbcUZVLUYsNiVRImFGJ0Y9RkBGW3BGUkZbcC1GLDYlUSJiRidGPUZARklGSS8lJW9wZW5HUSJ8ZnJGJy8lJmNsb3NlR1EifGhyRidGSUYrRkk=

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IiIi

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Valores Expl\303\255citos para LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JC1JJm1vdmVyR0YkNiYtRjA2JC1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y+USdub3JtYWxGJy1JI21vR0YkNi1RLSZjb25qdWdhdGUwO0YnRkAvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkgvJSlzdHJldGNoeUdGPC8lKnN5bW1ldHJpY0dGSC8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZILyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGV0ZTL0krbXNlbWFudGljc0dGJFEqY29uanVnYXRlRidGQEYrRkA=, LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JC1JJm1vdmVyR0YkNiYtRjA2JC1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y+USdub3JtYWxGJy1JI21vR0YkNi1RLSZjb25qdWdhdGUwO0YnRkAvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkgvJSlzdHJldGNoeUdGPC8lKnN5bW1ldHJpY0dGSC8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZILyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGV0ZTL0krbXNlbWFudGljc0dGJFEqY29uanVnYXRlRidGQEYrRkA=, and LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JC1JJm1vdmVyR0YkNiYtRjA2JC1GLDYlUSJ6RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y+USdub3JtYWxGJy1JI21vR0YkNi1RLSZjb25qdWdhdGUwO0YnRkAvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkgvJSlzdHJldGNoeUdGPC8lKnN5bW1ldHJpY0dGSC8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZILyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGV0ZTL0krbXNlbWFudGljc0dGJFEqY29uanVnYXRlRidGQEYrRkA=

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NiUjIihSaEoqIilLJD1cIkYjIyIoVnhdJCInI3l0Jg==

LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNictSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JUYrLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRitGMkY1

Tabela 37 Centro de massa de uma regi\303\243o espacial em coordenadas Cartesianas

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O modelo de tarefa conserta a ordem de integra\303\247\303\243o,mas o comando CenterOfMass ir\303\241 acertar qualquer uma das cinco ordens de integra\303\247\303\243o sobre uma regi\303\243o em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjNGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 3" layout="Heading 3">Coordenadas Cil\303\255ndricas</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Para encontrar o centro de massa da regi\303\243o

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

cuja densidade \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJW1zdXBHRiQ2JS1GLDYlUSJ6RicvRjBRJXRydWVGJy9GM1EnaXRhbGljRictRiM2JS1JI21uR0YkNiRRIjJGJ0YyRlJGVC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGMg==, use o modelo de tarefa na Tabela 38.

Ferramentas\342\211\273Tarefas\342\211\273Pesquisar: Multivariate Calculus\342\211\273Center of Mass\342\211\273Cylindrical

Centro de Massa para Regi\303\265es 3D em Coordenadas Cil\303\255ndricas

Densidade:

LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNictSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2J0YrLUkobWFjdGlvbkdGJDYkLUYwNiUtSSVtc3VwR0YkNiUtRiw2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GMDYlLUkjbW5HRiQ2JFEiMkYnL0ZBUSdub3JtYWxGJ0Y9RkAvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRidGSS8lK2FjdGlvbnR5cGVHUTttYXBsZXNvZnQ6dGFzay1wbGFjZWhvbGRlckYnRitGTkZJRitGTkZJ

KiQpSSJ6RzYiIiIjIiIi

Regi\303\243o: 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IiIi

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IiIh

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IiIi

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IiIh

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LCQqJiMiIiIiIiRGJUkjUGlHJSpwcm90ZWN0ZWRHRiVGJQ==

Momentos \303\267 Massa:

Integral Inerte - LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2Ki1GLDYlUSNkekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIn5GJ0Y1RjgvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkEvJSlzdHJldGNoeUdGQS8lKnN5bW1ldHJpY0dGQS8lKGxhcmdlb3BHRkEvJS5tb3ZhYmxlbGltaXRzR0ZBLyUnYWNjZW50R0ZBLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUC1GLDYlUSNkckYnRjVGOEY7LUYsNiVRImRGJ0Y1RjgtRiw2JVEnJiM5NTI7RicvRjZGQS9GOVEnbm9ybWFsRidGNUY4RitGZ24=

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Valores Expl\303\255citos para LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JC1JJm1vdmVyR0YkNiYtRjA2JC1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y+USdub3JtYWxGJy1JI21vR0YkNi1RLSZjb25qdWdhdGUwO0YnRkAvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkgvJSlzdHJldGNoeUdGPC8lKnN5bW1ldHJpY0dGSC8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZILyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGV0ZTL0krbXNlbWFudGljc0dGJFEqY29uanVnYXRlRidGQEYrRkA=, LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JC1JJm1vdmVyR0YkNiYtRjA2JC1GLDYlUSgmdGhldGE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y9LUkjbW9HRiQ2LVEtJmNvbmp1Z2F0ZTA7RidGPS8lJmZlbmNlR0Y8LyUqc2VwYXJhdG9yR0Y8LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjwvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGPC8lJ2FjY2VudEdGPC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlVGUS9JK21zZW1hbnRpY3NHRiRRKmNvbmp1Z2F0ZUYnRj1GK0Y9, e LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JC1JJm1vdmVyR0YkNiYtRjA2JC1GLDYlUSJ6RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y+USdub3JtYWxGJy1JI21vR0YkNi1RLSZjb25qdWdhdGUwO0YnRkAvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkgvJSlzdHJldGNoeUdGPC8lKnN5bW1ldHJpY0dGSC8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZILyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGV0ZTL0krbXNlbWFudGljc0dGJFEqY29uanVnYXRlRidGQEYrRkA=, o centro de massadado em coordenadas cil\303\255ndricas:

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

NiUsJComIyIjOyIiKiIiIkkjUGlHJSpwcm90ZWN0ZWRHISIiRigsJComI0YoIiInRihGKUYoRigjIiIlIiIm

LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNictSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JUYrLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRitGMkY1

Tabela 38 Centro de massa de uma regi\303\243o espacial em coordenadas cil\303\255ndricas

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 3" layout="Heading 3">Coordenadas Esf\303\251ricas</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

To find the center of mass of the region

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JKG1mZW5jZWRHRiQ2Ji1GIzY4LUZQNiQtRiM2KC1GLDYlUScmIzk2MTtGJy9GMEY9RjktRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTlEsMC4zMzMzMzMzZW1GJy1GLDYlUScmIzk2NjtGJ0ZlbkY5RmZuLUYsNiVRJyYjOTUyO0YnRmVuRjlGOUY5LUY2Ni1RIjpGJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUkjbW5HRiQ2JFEiMEYnRjktRjY2LVEmJmxlcTtGJ0Y5RjtGPkZARkJGREZGRkhGSkZNRlhGW3AtRmhvNiRRIjFGJ0Y5RmZuRmdvRltwRl5vRltwLUkmbWZyYWNHRiQ2KC1GLDYlUScmIzk2MDtGJ0ZlbkY5LUYjNiUtRmhvNiRRIjZGJ0Y5Ri9GMi8lLmxpbmV0aGlja25lc3NHRmBwLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmBxLyUpYmV2ZWxsZWRHRj1GZm5GZ29GW3BGYW9GW3AtRmhvNiRRIjJGJ0Y5LUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkhGam4vRk5GW29GZHBGOUY5LyUlb3BlbkdRInxmckYnLyUmY2xvc2VHUSJ8aHJGJ0Y5

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

whose density is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy, use the task template in Table 39.

Tools\342\211\273Tasks\342\211\273Browse: Multivariate Calculus\342\211\273Center of Mass\342\211\273Spherical

Centro de Massa para uma Regi\303\243o 3D em Coordenadas Esf\303\251ricas

(LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JC1GLDYlUScmIzk2NjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjhGK0Y4 \303\251 o \303\242ngulo azimutal)

Densidade:

LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNictSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2J0YrLUYwNidGKy1JKG1hY3Rpb25HRiQ2JC1GMDYmLUYsNiZRJiZyaG87RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZm9yZWdyb3VuZEdRK1s5NSwwLDIxN11GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStleGVjdXRhYmxlR0Y+LyUscGxhY2Vob2xkZXJHUSV0cnVlRidGQi8lK2FjdGlvbnR5cGVHUTttYXBsZXNvZnQ6dGFzay1wbGFjZWhvbGRlckYnRitGRUZCRitGRUZCRitGRUZC

SSRyaG9HNiI=

Regi\303\243o: LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JC1JKG1mZW5jZWRHRiQ2Ji1GMDY4LUklbXN1YkdGJDYlLUYsNiVRJiZyaG87RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GMDYkLUkjbW5HRiQ2JFEiMUYnRkBGQC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUYzNiQtRjA2Ji1GLDYlUScmIzk2NjtGJ0Y9RkAtSSNtb0dGJDYtUSIsRidGQC8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRJyYjOTUyO0YnRj1GQEZARkAtRlQ2LVEmJmxlcTtGJ0ZARlcvRlpGP0ZmbkZobkZqbkZcb0Zeby9GYW9RLDAuMjc3Nzc3OGVtRicvRmRvRl5wLUYsNiVRJyYjOTYxO0YnRj1GQEZpby1GODYlRjotRjA2JC1GRjYkUSIyRidGQEZARklGTEZTLUY4NiVGUEZDRkktRjM2JC1GMDYkRmZvRkBGQEZpb0ZQRmlvLUY4NiVGUEZlcEZJRlxxRlMtRiw2JVEiYUYnL0Y+RmVuL0ZBUSdpdGFsaWNGJ0Zpb0Zmb0Zpby1GLDYlUSJiRidGZXFGZnFGQEZALyUlb3BlbkdRInxmckYnLyUmY2xvc2VHUSJ8aHJGJ0ZARitGQA==

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

LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNictSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2J0YrLUYwNidGKy1JKG1hY3Rpb25HRiQ2JC1GMDYmLUkjbW5HRiQ2JVEiMEYnLyUrZm9yZWdyb3VuZEdRK1s5NSwwLDIxN11GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLHBsYWNlaG9sZGVyR1EldHJ1ZUYnRkAvJSthY3Rpb250eXBlR1E7bWFwbGVzb2Z0OnRhc2stcGxhY2Vob2xkZXJGJ0YrRkNGQEYrRkNGQEYrRkNGQA==

IiIh

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

LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNictSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2J0YrLUkobWFjdGlvbkdGJDYkLUYwNihGKy1GMDYlLUkjbW5HRiQ2JVEiMUYnLyUrZm9yZWdyb3VuZEdRK1s5NSwwLDIxN11GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJ0ZARitGQy8lLHBsYWNlaG9sZGVyR1EldHJ1ZUYnRkAvJSthY3Rpb250eXBlR1E7bWFwbGVzb2Z0OnRhc2stcGxhY2Vob2xkZXJGJ0YrRkNGQEYrRkNGQA==

IiIi

LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JEYrLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0YrRjI=

LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JS1JJW1zdWJHRiQ2JS1GLDYlUScmIzk2NjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYwNiQtSSNtbkdGJDYkUSIxRidGO0Y7LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiQtRjA2JC1GLDYlUScmIzk1MjtGJ0Y4RjtGO0Y7RjtGK0Y7

LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNictSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2J0YrLUYwNidGKy1JKG1hY3Rpb25HRiQ2JC1GMDYmLUkjbW5HRiQ2JVEiMEYnLyUrZm9yZWdyb3VuZEdRK1s5NSwwLDIxN11GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLHBsYWNlaG9sZGVyR1EldHJ1ZUYnRkAvJSthY3Rpb250eXBlR1E7bWFwbGVzb2Z0OnRhc2stcGxhY2Vob2xkZXJGJ0YrRkNGQEYrRkNGQEYrRkNGQA==

IiIh

LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JEYrLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0YrRjI=

LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JS1JJW1zdWJHRiQ2JS1GLDYlUScmIzk2NjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYwNiQtSSNtbkdGJDYkUSIyRidGO0Y7LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiQtRjA2JC1GLDYlUScmIzk1MjtGJ0Y4RjtGO0Y7RjtGK0Y7

LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNictSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2J0YrLUYwNidGKy1JKG1hY3Rpb25HRiQ2JC1GMDYmLUkmbWZyYWNHRiQ2KC1GLDYmUSUmcGk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZm9yZWdyb3VuZEdRK1s5NSwwLDIxN11GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRjA2JS1JI21uR0YkNiVRIjZGJ0ZCRkUvJStleGVjdXRhYmxlR0ZBRkUvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRlUvJSliZXZlbGxlZEdGQUZOLyUscGxhY2Vob2xkZXJHUSV0cnVlRidGRS8lK2FjdGlvbnR5cGVHUTttYXBsZXNvZnQ6dGFzay1wbGFjZWhvbGRlckYnRitGTkZFRitGTkZFRitGTkZF

LCQqJiMiIiIiIidGJUkjUGlHJSpwcm90ZWN0ZWRHRiVGJQ==

LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JEYrLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0YrRjI=

IiIh

LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JEYrLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0YrRjI=

LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiktSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JUYrLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJ0YrLUYwNiYtSSNtbkdGJDYkUSIyRicvRjZRJ25vcm1hbEYnLUkjbW9HRiQ2LVEifkYnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkYvJSlzdHJldGNoeUdGRi8lKnN5bW1ldHJpY0dGRi8lKGxhcmdlb3BHRkYvJS5tb3ZhYmxlbGltaXRzR0ZGLyUnYWNjZW50R0ZGLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVS1GLDYlUScmIzk2MDtGJy9GM0ZGRj5GPkYrLyUrZXhlY3V0YWJsZUdGRkY+

LCQqJiIiIyIiIkkjUGlHJSpwcm90ZWN0ZWRHRiVGJQ==

LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JEYrLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0YrRjI=

Momentos \303\267 Massa:

Integral Inerte- LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2Ky1GLDYlUSJkRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYsNiVRJyYjOTYxO0YnL0Y2USZmYWxzZUYnL0Y5USdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRj8vJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVkYyLUYsNiVRJyYjOTY2O0YnRj5GQEZCRjItRiw2JVEnJiM5NTI7RidGPkZARkBGK0ZA

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

NiUqJi1JJEludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLUYlNiQtRiU2JCooKS1JJHNpbkdGJjYjSSRwaGlHRikiIiMiIiItSSRjb3NHRiY2I0kmdGhldGFHRilGNilJJHJob0dGKSIiJUY2L0Y6OyIiISwkKiZGNUY2SSNQaUdGJ0Y2RjYvRjQ7RkAsJComI0Y2IiInRjZGQ0Y2RjYvRjw7RkBGNkY2LUYlNiQtRiU2JC1GJTYkKiYpRjwiIiRGNkYxRjZGPkZERkohIiIqJi1GJTYkLUYlNiQtRiU2JCooRjBGNi1GMkY5RjZGO0Y2Rj5GREZKRjZGTEZVKiYtRiU2JC1GJTYkLUYlNiQqKC1GOEYzRjZGO0Y2RjFGNkY+RkRGSkY2RkxGVQ==

LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JEYrLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0YrRjI=

Valores Expl\303\255citos para LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JS1JJm1vdmVyR0YkNiYtRjA2JC1GLDYlUSYmcmhvO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGPS1JI21vR0YkNi1RLSZjb25qdWdhdGUwO0YnRj0vJSZmZW5jZUdGPC8lKnNlcGFyYXRvckdGPC8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y8LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjwvJSdhY2NlbnRHRjwvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZVRlEvSSttc2VtYW50aWNzR0YkUSpjb25qdWdhdGVGJ0YrRj1GK0Y9,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 e LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JC1JJm1vdmVyR0YkNiYtRjA2JC1GLDYlUSgmdGhldGE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y9LUkjbW9HRiQ2LVEtJmNvbmp1Z2F0ZTA7RidGPS8lJmZlbmNlR0Y8LyUqc2VwYXJhdG9yR0Y8LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjwvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGPC8lJ2FjY2VudEdGPC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlVGUS9JK21zZW1hbnRpY3NHRiRRKmNvbmp1Z2F0ZUYnRj1GK0Y9, o centro de massa dado em coordenadas esf\303\251ricas:

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

NiUsJCooIyIiIiIjP0YmSSNQaUclKnByb3RlY3RlZEdGJiwmKiYjRiYiIiNGJkYoRiZGJiooI0YmIiIlRiZGKEYmKSIiJEYsRiYhIiJGM0YmIiIhRjQ=

LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JEYrLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0YrRjI=

LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNictSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JUYrLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRitGMkY1

Tabela 39 Centro de massa de uma regi\303\243o espacial em coordenadas esf\303\251ricas

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=