C\303\241lculo IILi\303\247\303\243o 7: Aplica\303\247\303\265es da Integra\303\247\303\243o 5, Parte 2: Centr\303\263ides Primeiro momento, Teorema dos centr\303\263ides de Papus Quando o conceito de centro de gravidade ou centro de massa est\303\241 restrito a uma curva plana fechada, tem-se a id\303\251ia de "centr\303\263ide". O centr\303\263ide \303\251 o ponto de simetria de uma curva fechada. \303\211 tamb\303\251m o "centro de massa". Consideremos a parte da par\303\241bola descrita por 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 que encontra-se sobre o eixo x. Como trata-se de uma superf\303\255cie plana, a densidade \303\251 constante e n\303\243o precisa ser levada em conta. Al\303\251m disso, o centr\303\263ide \303\251 uma ponto geomentricamente definido, cuja localiza\303\247\303\243o \303\251 independente de coordenadas. Dividimos agora a \303\241rea da superf\303\255cie em faixas de comprimento LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSgmRGVsdGE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi5RMSZJbnZpc2libGVUaW1lcztGJy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiw2JVEieEYnL0Y1USV0cnVlRicvRjhRJ2l0YWxpY0YnRis= e medimos a dist\303\242ncia do centro da faixa, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRImlGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== , ao eixo y. Se aproximarmos a \303\241rea de cada faixa por 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, ent\303\243o tem-se que o primeiro momento \303\251 definido por 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 , onde n \303\251 o n\303\272mero de faixas. No limite quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSgmRGVsdGE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi5RMSZJbnZpc2libGVUaW1lcztGJy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiw2JVEieEYnL0Y1USV0cnVlRicvRjhRJ2l0YWxpY0YnRis=->0, este torna-se 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. O primeiro momento sobre o eixo y \303\251, portanto, 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 . Sendo assim, a coordenada x do centr\303\263ide \303\251 dada por 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. Para o c\303\241lculo da coordenada y do centr\303\263ide, deve-se observar que, dada uma faixa infinitesimalmente larga, o centr\303\263ide encontra-se na metade da dist\303\242ncia entre as extremidades superior e inferior da faixa. Portanto, 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 Ex. 1 Localizar o centr\303\263ide da \303\241rea plana contida entre a curva 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, o eixo y e a reta x = 3. restart:with(plots):

Warning, the name changecoords has been redefined

f:=x->-x^2+3*x;

NiM+JSJmR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0c2IiwmKiQpJSJ4RyIiIyIiIiEiIiomIiIkIiIiJSJ4RyIiIiIiIjYiNiI2Ig==

A:=int(f(x),x=0..3);

NiM+JSJBRyMiIioiIiM=

C[x]:=int(x*f(x),x=0..3)/A;

NiM+JiUiQ0c2IyUieEcjIiIkIiIj

C[y]:=int(f(x)^2,x=0..3)/(2*A);

NiM+JiUiQ0c2IyUieUcjIiIqIiM1

p1:=plot(f(x),x=0..3): p2:=pointplot([[C[x],C[y]]],symbol=circle): display(p1,p2);

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NiI=

O centr\303\263ide encontra-se na linha de simetria da \303\241rea da superf\303\255cie, como j\303\241 era de se esperar. Adicionamos a seguir uma pequena complica\303\247\303\243o. Ex. 2 Tra\303\247amos agora o gr\303\241fico de sin(x) e do c\303\255rculo centrado em ( 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) com raio 1 e localizamos o centr\303\263ide da figura resultande. restart: with(plots):

Warning, the name changecoords has been redefined

f:=x->sin(x);

NiM+JSJmRyUkc2luRw==

Eq1:=(x-Pi/2)^2+(y-1)^2=1;

NiM+JSRFcTFHLywmKiQpLCYlInhHIiIiKiYiIiMhIiIlI1BpRyIiIiEiIiIiIyIiIiIiIiokKSwmJSJ5RyIiIiIiIiEiIiIiIyIiIiIiIiIiIg==

Y:=solve(Eq1,y);

NiM+JSJZRzYkLCYiIiIiIiIqJiIiIyEiIiksKiIiJSIiIiomIiIlIiIiKSUieEciIiMiIiIhIiIqKCIiJSIiIiUieEciIiIlI1BpRyIiIiIiIiokKSUjUGlHIiIjIiIiISIiIyIiIiIiIyIiIiIiIiwmIiIiIiIiKiYiIiMhIiIpLCoiIiUiIiIqJiIiJSIiIiklInhHIiIjIiIiISIiKigiIiUiIiIlInhHIiIiJSNQaUciIiIiIiIqJCklI1BpRyIiIyIiIiEiIiMiIiIiIiMiIiIhIiI=

g:=unapply(Y[2],x);

NiM+JSJnR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0c2IiwmIiIiIiIiKiYjIiIiIiIjIiIiKiQpLCoiIiUiIiIqJiIiJSIiIiklInhHIiIjIiIiISIiKigiIiUiIiIlInhHIiIiJSNQaUciIiIiIiIqJCklI1BpRyIiIyIiIiEiIiMiIiIiIiMiIiIiIiIhIiI2IjYiNiI=

plot({f(x),g(x)},x=0..3,scaling=constrained);

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NiI=

Pode-se agora encontrar o centr\303\263ide da \303\241rea contida entre os dois gr\303\241ficos. Em primeiro lugar, \303\251 preciso observar que a \303\241rea de interesse \303\251 descrita por f(x) - g(x). O centro de cada faixa \303\251 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 acima do eixo x. O comprimento de cada faixa \303\251 f(x) - g(x) . Sendo assim, as f\303\263rmulas para o centr\303\263ide requerem uma modifica\303\247\303\243o: 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 = 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 Inicialmente, \303\251 preciso encontrar os pontos de interse\303\247\303\243o das duas curvas para o c\303\241lculo dos limites de integra\303\247\303\243o. x1:=fsolve(f(x)=g(x),x=0..1);

NiM+JSN4MUckIislb3NeXichIzU=

x2:= fsolve(f(x)=g(x),x=2..3);

NiM+JSN4MkckIismUXYrXCMhIio=

A:=int(f(x)-g(x),x=x1..x2);

NiM+JSJBRyQiK1gtJil6NyEiKg==

C[x]:=int(x*(f(x)-g(x)),x=x1..x2)/A;

NiM+JiUiQ0c2IyUieEckIitFanpxOiEiKg==

C[y]:=int((f(x)^2-g(x)^2),x=x1..x2)/(2*A);

NiM+JiUiQ0c2IyUieUckIiswZV1zXSEjNQ==

p1:=plot({f(x),g(x)},x=0..3,scaling=constrained, labels=[x,y]): p2:=pointplot([[C[x],C[y]]],symbol=circle): display(p1,p2);

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Ig==

A resposta est\303\241 visivelmente correta. Teorema de Pappus Dada uma curva fechada de \303\241rea A, per\303\255metro P e centr\303\263ide {LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRInhGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIixGJy9GNlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGSC8lKnN5bW1ldHJpY0dGSC8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZILyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JUYuLUYjNiMtRi82JVEieUYnRjJGNUY9}, e uma reta externa \303\240 curva fechada cuja dist\303\242ncia do centr\303\263ide \303\251 d, pode-se girar a curva plana em torno desta reta de forma a obter-se um s\303\263lido de revolu\303\247\303\243o. O volume do s\303\263lido \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJWRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYjNiktSSNtbkdGJDYkUSIyRidGPi1GOzYuUTEmSW52aXNpYmxlVGltZXM7RicvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnRj5GQEZDRkVGR0ZJRktGTS9GUFEmMC4wZW1GJy9GU0Zbby1GLDYlUSNQaUYnL0Y1RkJGPkZaLUYsNiVRImRGJ0Y0RjdGWi1GLDYlUSJBRidGNEY3RitGKw== , e a \303\241rea da superf\303\255cie \303\251 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. Ex. 3 Dada a elipse: 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 , encontrar a \303\241rea da superf\303\255cie do s\303\263lido de revolu\303\247\303\243o em torno do eixo x. Deve-se girar a elipse em torno do eixo x. restart: with(plots):

Warning, the name changecoords has been redefined

Eq1:=(x-5)^2/9+(y-7)^2/4=1;

NiM+JSRFcTFHLywmKiYiIiohIiIpLCYlInhHIiIiIiImISIiIiIjIiIiIiIiKiYiIiUhIiIpLCYlInlHIiIiIiIoISIiIiIjIiIiIiIiIiIi

E:=solve(Eq1,y);

NiM+JSJFRzYkLCYiIigiIiIqKCIiIyIiIiIiJCEiIiksKCIjOyEiIiokKSUieEciIiMiIiIhIiIqJiIjNSIiIiUieEciIiIiIiIjIiIiIiIjIiIiIiIiLCYiIigiIiIqKCIiIyIiIiIiJCEiIiksKCIjOyEiIiokKSUieEciIiMiIiIhIiIqJiIjNSIiIiUieEciIiIiIiIjIiIiIiIjIiIiISIi

plot3d plot3d({[x,E[1]*cos(theta),E[1]*sin(theta)],[x,E[2]*cos(theta),E[2]*sin(theta)]} ,x=2..8,theta=0..2*Pi,scaling=constrained,axes=boxed,color=theta);

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Como, por simetria, sabe-se que o centr\303\263ide da elipse encontra-se em (5,7) e a \303\241rea da elipse \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEjUGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn (semi-eixo maior)x(semi-eixo menor), tem-se, imediatamente, o volume do s\303\263lido resultante: Volume:=2*Pi*7*Pi*3*2;

NiM+JSdWb2x1bWVHLCQqJiIjJSkiIiIpJSNQaUciIiMiIiIiIiI=

Ex. 4 Dado o arco parab\303\263lico 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, e a reta y = 10, deseja-se saber qual o volume do s\303\263lido de revolu\303\247\303\243o obtido pela rota\303\247\303\243o desta \303\241rea em torno de uma reta paralela ao eixo x e a uma dist\303\242ncia de 5 unidades do menor ponto no arco. Para este problema basta conhecer a coordenada y do centr\303\263ide. f:=x->3*x^2+4*x+5;

NiM+JSJmR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0c2IiwoKiYiIiQiIiIpJSJ4RyIiIyIiIiIiIiomIiIlIiIiJSJ4RyIiIiIiIiIiJiIiIjYiNiI2Ig==

plot({f(x),10},x=-4..2);

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NiI=

Inicialmente, \303\251 preciso conhecer os limites de integra\303\247\303\243o. x1:=fsolve(f(x)=10,x=-3..-2); x2:=fsolve(f(x)=10,x=0..1);

NiM+JSN4MUckISsiKUhqPkAhIio=NiM+JSN4MkckIit5aypIJ3khIzU=

Pode-se ent\303\243o calcular a coordenada y do centr\303\263ide. A:=int((10-f(x)),x=x1..x2);

NiM+JSJBRyQiK0skXHBBIiEiKQ==

c:=int((10^2-f(x)^2)/2,x=x1..x2)/A;

NiM+JSJjRyQiK29tbW11ISIq

\303\211 precisso conhecer tamb\303\251m a dist\303\242ncia do centr\303\263ide em rela\303\247\303\243o ao menor ponto da curva, que ocorre em 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. h:=c-f(-4.0/6);

NiM+JSJoRyQiKywrKytRISIq

O volume procurado \303\251: Volume:=evalf(2*Pi*(5+h)*A);

NiM+JSdWb2x1bWVHJCIrOD8wJXknISIo

Apresentamos a seguir o formato do s\303\263lido. plot3d({[x,f(x)*cos(theta),f(x)*sin(theta)],[x,10*cos(theta),10*sin(theta)]},x=x1..x2, theta=0..2*Pi,view=[-2..2,-10..10,-10..10],style=wireframe,color=theta/10,axes=normal);

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