C\303\241lculo II Li\303\247\303\243o 13a: Integra\303\247\303\243o Num\303\251rica-Parte 1
M\303\251todo de Euler. Deriva\303\247\303\243o pela regra do trap\303\251zio e de Simpson e sua utilidade. simpson, trap\303\251zio
A maior parte das fun\303\247\303\265es n\303\243o possui uma integral exata. Por isso, s\303\243o utilizadas v\303\241rias t\303\251cnicas para o c\303\241lculo do valor das integrais definidas de tais fun\303\247\303\265es. H\303\241 uma grande variedade de m\303\251todos para obten\303\247\303\243o de solu\303\247\303\265es num\303\251ricas para integrais definidas. Consideraremos aqui alguns destes m\303\251todos; em particular, o m\303\251todo de Euler e o m\303\251todo da quadratura.
M\303\251todo de Euler
Dada uma derivada, 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 , juntamente com uma condi\303\247\303\243o inicial, 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 , o m\303\251todo de Euler gera uma sequ\303\252ncia de pontos que aproximam y (x ), a come\303\247ar pelo ponto inicial dado. Para isso, \303\251 utilizado o valor de g '(x ) a cada ponto.
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JJW1zdWJHRiQ2JS1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiZGKy1GIzYmLUYsNiVRIm5GJ0Y3RjotSSNtb0dGJDYtUSIrRicvRjtRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZMLyUpc3RyZXRjaHlHRkwvJSpzeW1tZXRyaWNHRkwvJShsYXJnZW9wR0ZMLyUubW92YWJsZWxpbWl0c0dGTC8lJ2FjY2VudEdGTC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRmVuLUkjbW5HRiQ2JFEiMUYnRkhGSEYrRkgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GRTYtUSI9RidGSEZKRk1GT0ZRRlNGVUZXL0ZaUSwwLjI3Nzc3NzhlbUYnL0ZnbkZjby1GMjYlRjQtRiM2JEZBRkhGXG9GSEYrRkg= + g' ( 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 )LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUScmIzkxNjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnRjcvJSZmZW5jZUdGNi8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0Y2LyUobGFyZ2VvcEdGNi8lLm1vdmFibGVsaW1pdHNHRjYvJSdhY2NlbnRHRjYvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZOLUYsNiVRInhGJy9GNVEldHJ1ZUYnL0Y4USdpdGFsaWNGJ0Y3RitGNw== , onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5MTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy x \303\251 um pequeno intervalo escolhido arbitrariamente, digamos 0.01 unidades. O que se faz n\303\243o \303\251 nada al\303\251m de aproximar cada LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRIm5GJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== sucessivo recursivamente, utilizando para isso os m\303\251todos do c\303\241lculo diferencial.
Ex. 15.1
Dada a condi\303\247\303\243o inicial, y (1)=1, encontrar a anti-derivada de 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 .
restart: with(plots):
df:=x->sin(x)^8;
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNkZkdGKGYqNiNJInhHRihGKDYkSSlvcGVyYXRvckdGKEkmYXJyb3dHRihGKCokLUkkc2luR0YlNiM5JCIiKUYoRihGKDcjRi4=
y[n+1]:=y[n]+df(x[1]+n*delta)*(delta);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkieUdGKDYjLCZJIm5HRigiIiJGMkYyLCYmRi42I0YxRjIqJi1JJHNpbkdGJTYjLCYqJkkmZGVsdGFHRihGMkYxRjJGMiZJInhHRig2I0YyRjIiIilGPEYyRjI3I0Yz
x[1]:=1: y[1]:=1: delta:=0.01:
for n to 100 do y[n+1]:=y[n]+df(x[1]+n*delta)*(delta): x[n+1]:=x[1]+n*delta: od:
S:=[seq([x[k],y[k]],k=1..101)]:
pointplot(S,symbol=circle);
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NiI=
Deseja-se encontrar y (x ) para qualquer valor de x entre 1 e 2. Em particular, calculamos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1JI21uR0YkNiRRJTEuNzRGJ0Y+Rj5GPkY+RitGPg== .
y[74];
JCIrQmdxXDohIio=
Neste caso, o Maple facilmente fornece uma solu\303\247\303\243o exata.
G:=int(df(x),x);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJHR0YoLCYqJiwqKiQtSSRzaW5HRiU2I0kieEdGKCIiKCIiIiokRjIiIiYjRjYiIicqJEYyIiIkIyIjTiIjQ0YyI0Y/IiM7RjctSSRjb3NHRiVGNEY3IyEiIiIiKUY1I0Y/IiRHIjcjRi4=
G1:=evalf(subs(x=1,G));
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHMUdGKCQiKikqejxxJCEjNTcjRi4=
K:=solve(G1+K=1,K);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoJCIrLT8jKUgnKiEjNTcjRi4=
G:=G+K:
Observemos agora os gr\303\241ficos simult\303\242neos da solu\303\247\303\243o exata e da solu\303\247\303\243o num\303\251rica calculada pelo m\303\251todo de Euler.
P1:=pointplot(S):
P2:=plot(G,x=1..2):
display(P1,P2);
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Os dois gr\303\241ficos encontram-se sobrepostos. A precis\303\243o do m\303\251todo de Euler depende do tamanho dos intervalos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5MTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy x ; quanto menor o intervalo, mais preciso \303\251 o resultado.
Quadratura
Pelo gr\303\241fico, fica claro que uma aproxima\303\247\303\243o razo\303\241vel para a \303\241rea sob a curva 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 , de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GPkYrRj4= a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiNEYnRj5GPkYrRj4= , pode ser obtida pelo c\303\241lculo das \303\241reas dos trap\303\251zios inscritos. Uma aproxima\303\247\303\243o ainda melhor pode ser obtida com a redu\303\247\303\243o do intervalo pela metade, entre os pontos onde f \303\251 calculada. O leitor, sem d\303\272vida, j\303\241 deve estar bastante familiar com a express\303\243o para a \303\241rea do trap\303\251zio. Por\303\251m, a fim de estabelecermos e ilustrarmos algumas id\303\251ias, utilizaremos os m\303\251todos do c\303\241lculo. Seja h o intervalo entre os valore de x .
restart:with(plots):
f:=x->x^2:
P1:=plot(f,1..5,color=blue,thickness=2):
t1:=[[1,0],[1,f(1)],[2,f(2)],[2,0]]:
t2:=[[2,0],[2,f(2)],[3,f(3)],[3,0]]:
t3:=[[3,0],[3,f(3)],[4,f(4)],[4,0]]:
t4:=[[4,0],[4,f(4)],[5,f(5)],[5,0]]:
P2:=polygonplot({t1,t2,t3,t4}):
display(P1,P2);
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Exercitaremos agora algumas habilidades de c\303\241lculo simb\303\263lico do Maple . Inicialmente, calcularemos a equa\303\247\303\243o para a secante que une dois pontos, 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 e 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 . Note que existe uma distin\303\247\303\243o entre as letras mai\303\272sculas X e Y , usadas como vari\303\241veis, e as letras min\303\272sculas x e y , usadas para identificar pontos espec\303\255ficos.
Eq1:=(y[1]-y[0])/(x[1]-x[0])=(Y-y[0])/(X-x[0]);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRFcTFHRigvKiYsJiZJInlHRig2IyIiIkY0JkYyNiMiIiEhIiJGNCwmJkkieEdGKEYzRjQmRjtGNkY4RjgqJiwmSSJZR0YoRjRGNUY4RjQsJkkiWEdGKEY0RjxGOEY4NyNGLg==
secant:=solve(Eq1,Y);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdzZWNhbnRHRigqJiwqKiZJIlhHRigiIiImSSJ5R0YoNiMiIiFGMkYyKiZGMUYyJkY0NiNGMkYyISIiKiYmSSJ4R0YoRjVGMkY4RjJGMiomJkY9RjlGMkYzRjJGOkYyLCZGP0Y6RjxGMkY6NyNGLg==
Pode-se agora encontrar a \303\241rea sob a secante por integra\303\247\303\243o.
integral:=int(secant, X=x[0]..x[1]);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlpbnRlZ3JhbEdGKCwmKigsJiZJInlHRig2IyIiISIiIiZGMjYjRjUhIiJGNSwmJkkieEdGKEY3RjgmRjtGM0Y1RjgsJiokRjwiIiNGOCokRjpGP0Y1RjUjRjVGPyooLCYqJkY8RjVGNkY1RjUqJkY6RjVGMUY1RjhGNUY5RjgsJkY6RjVGPEY4RjVGNTcjRi4=
O resultado acima ainda pode ser simplificado!
Area:=factor(integral);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVBcmVhR0YoLCQqJiwmJkkieUdGKDYjIiIhIiIiJkYyNiNGNUY1RjUsJiZJInhHRihGNyEiIiZGOkYzRjVGNSNGOyIiIzcjRi4=
Ainda \303\251 poss\303\255vel simplificar o resultado com a substitui\303\247\303\243o x1=x0+h !
Area:=subs(x[1]=x[0]+h ,Area);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVBcmVhR0YoLCQqJiwmJkkieUdGKDYjIiIhIiIiJkYyNiNGNUY1RjVJImhHRihGNSNGNSIiIzcjRi4=
Obt\303\251m-se assim a conhecida f\303\263rmula para a \303\241rea do trap\303\251zio. Pela adi\303\247\303\243o de n trap\303\251zio sucessivos, obt\303\251m-se uma f\303\263ruma aproximada, conhecida como regra do trap\303\251zio.
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNiotSShtc3Vic3VwR0YkNictSSNtb0dGJDYvUSgmIzg3NDc7RicvJStmb3JlZ3JvdW5kR1EuWzE0NCwxNDQsMTQ0XUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy9JK21zZW1hbnRpY3NHRiRRJmluZXJ0RicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVC1JJW1zdWJHRiQ2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRj5RJ2l0YWxpY0YnLUYjNiQtSSNtbkdGJDYkUSIwRidGPUY9LyUvc3Vic2NyaXB0c2hpZnRHRmFvLUZYNiVGWi1GIzYkLUYsNiVRIm5GJ0ZnbkZqbkY9RmJvLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjJGJ0Zib0YrLUYjNiYtRiw2JVEieUYnRmduRmpuLUY3Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRj1GQ0ZGRkhGSkZMRk5GUEZSRlUtSShtZmVuY2VkR0YkNiQtRiM2JEZaRj1GPUY9RistSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuM2VtRicvJSZkZXB0aEdGYHEvJSpsaW5lYnJlYWtHUSVhdXRvRictRjc2L1EwJkRpZmZlcmVudGlhbEQ7RidGOkY9RkBGQ0ZGRkhGSkZMRk5GUEZSRlVGWkY9LUY3Ni1RIj1GJ0Y9RkNGRkZIRkpGTEZORlAvRlNRLDAuMjc3Nzc3OGVtRicvRlZGYHItSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEiaEYnRmduRmpuRj0tRiM2JC1GX282JEZdcEY9Rj0vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmNzLyUpYmV2ZWxsZWRHRkVGPUYrRj0= (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHRj1GPg== 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 +LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUklbXN1YkdGJDYlLUYsNiVRInlGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRictRiM2JEYxRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1RitGNQ== +...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 +LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRIm5GJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== )
Pode-se fazer ainda melhor; em vez de aproximar a curvar com uma s\303\251rie de retas, pode-se utilizar curvas parab\303\263licas.
Para isso, deve-se selecionar tr\303\252s pontos, ( LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHRj0tSSNtb0dGJDYtUSIsRidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlLUYvNiVRInlGJ0YyRjVGOEZARj4= ) , (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSIsRidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0ZJLyUqc3ltbWV0cmljR0ZJLyUobGFyZ2VvcEdGSS8lLm1vdmFibGVsaW1pdHNHRkkvJSdhY2NlbnRHRkkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlLUYvNiVRInlGJ0YyRjVGOEZARj4= ) e (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSIsRidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0ZJLyUqc3ltbWV0cmljR0ZJLyUobGFyZ2VvcEdGSS8lLm1vdmFibGVsaW1pdHNHRkkvJSdhY2NlbnRHRkkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlLUYvNiVRInlGJ0YyRjVGOEZARj4= ), e encontrar uma par\303\241bola entre estes pontos.
restart:
for k from 0 to 2 do Eq||k:=a*x[k]^2+b*x[k]+c=y[k] od;
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRFcTBHRigvLCgqJkkiYUdGKCIiIiZJInhHRig2IyIiISIiI0YyKiZJImJHRihGMkYzRjJGMkkiY0dGKEYyJkkieUdGKEY1NyNGLg== LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRFcTFHRigvLCgqJkkiYUdGKCIiIiZJInhHRig2I0YyIiIjRjIqJkkiYkdGKEYyRjNGMkYySSJjR0YoRjImSSJ5R0YoRjU3I0Yu LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRFcTJHRigvLCgqJkkiYUdGKCIiIiZJInhHRig2IyIiI0Y2RjIqJkkiYkdGKEYyRjNGMkYySSJjR0YoRjImSSJ5R0YoRjU3I0Yu
solve({Eq0,Eq1,Eq2},{a,b,c});
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
assign(%):
parabola:=a*X^2+b*X+c;
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
Agora, basta integrar a \303\241rea sob a par\303\241bola!
integral:=int(parabola,X=x[0]..x[2]);
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
Faremos a seguir mais algumas substitui\303\247\303\265es! h \303\251 a dist\303\242ncia entre valores sucessivos de x , e corresponde a .
integral:=subs({x[1]=x[0]+h, x[2]=x[0]+2*h},integral);
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
Area:=expand(integral);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVBcmVhR0YoLCgqJiZJInlHRig2IyIiISIiIkkiaEdGKEY0I0Y0IiIkKiYmRjE2IyIiI0Y0RjVGNEY2KiYmRjE2I0Y0RjRGNUY0IyIiJUY3NyNGLg==
Area:=factor(Area);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVBcmVhR0YoLCQqJkkiaEdGKCIiIiwoJkkieUdGKDYjIiIhRjEmRjQ2IyIiI0YxJkY0NiNGMSIiJUYxI0YxIiIkNyNGLg==
Ao adicionarmos uma sequ\303\252ncia destes segmentos, obt\303\251m-se:
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 (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHRj1GPg== 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 +LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUklbXN1YkdGJDYlLUYsNiVRInlGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRictRiM2JEYxRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1RitGNQ== +LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUklbXN1YkdGJDYlLUYsNiVRInlGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRictRiM2JC1GMjYkUSIzRidGNUY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNUYrRjU= +...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 +LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRIm5GJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== )
Esta \303\251 a regra de Simpson. Esta t\303\251cnica requer um n\303\272mero \303\255mpar de pontos. A regra do trap\303\251zio e de Simpson s\303\243o exemplos de f\303\263rmulas de quadratura ou f\303\263rmul\303\241s de Newton-Cotes. F\303\263rmulas de quadratura de ordem superior podem ser obtidas de maneira similar. A fun\303\247\303\243o polim\303\263rfica ApproximateInt implementa tais m\303\251todos.
Ex. 15.2
Para o exemplo, 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 , tem-se os dados a seguir:
x 2.0 2.2 2.4 2.6 2.8 3.0 3.2 y(x) 4.0 5.666695779 8.175361775 11.99307928 17.86662356 27.0 41.35042053
Pela regra do trap\303\251zio obt\303\251m-se
Trap:=1/2*.2*(4+2*5.666695779+2*8.175361775+2*11.99307928+2*17.86662358+2*27.000+41.35042053);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVUcmFwR0YoJCIrOCVSdic9ISIpNyNGLg==
Pela regra de Simpson:
Simp:=1/3*.2*(4+4*5.666695779+2*8.175361775+4*11.99307928+2*17.86662358+4*27.000+41.35042053);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVTaW1wR0YoJCIrVioqW1M9ISIpNyNGLg==
Do Maple , tem-se:
with(Student[Calculus1]):
Com a regra do trap\303\251zio:
ApproximateInt(x^x,2.0..3.2,method=newtoncotes[1],output=plot,functionoptions=[color=yellow,thickness=3],color=black);
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""-%%ROOTG6'-%)BOUNDS_XG6#$"$q%!""-%)BOUNDS_YG6#$"$I$!""-%-BOUNDS_WIDTHG6#$"%SL!""-%.BOUNDS_HEIGHTG6#$"%]>!""-%)CHILDRENG6"-%+ANNOTATIONG6'-%)BOUNDS_XG6#$""!!""-%)BOUNDS_YG6#$""!!""-%-BOUNDS_WIDTHG6#$"%+S!""-%.BOUNDS_HEIGHTG6#$"%+S!""-%)CHILDRENG6"G6"
Com a regra de Simpson:
ApproximateInt(x^x,2.0..3.2,method=newtoncotes[2],output=plot,functionoptions=[color=yellow,thickness=3],color=black);
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Utilizando os algor\303\255timos de integra\303\247\303\243o num\303\251rica do Maple :
evalf(int(x^x,x=2..3.2));
JCIra1wzUz0hIik=
O Maple possui ainda duas fun\303\247\303\265es, trapezoid e simpson student .
A exist\303\252ncia destas fun\303\247\303\265es n\303\243o elimina a necessidade em se conhecer como s\303\243o usadas estas e outras regras de quadratura, como mostram os exemplos a seguir.
Ex. 15.3
Dada uma fun\303\247\303\243o na forma de uma tabela de medidas, resultado de algum processo cont\303\255nuo, deseja-se integrar a fun\303\247\303\243o representada por estes pontos, entre os limites das medidas.
x 1.0 1.25 1.5 1.75 2.0 f(x) 3.91 4.11 4.32 4.54 4.77
As fun\303\247\303\265es do Maple n\303\243o s\303\243o de grande ajuda neste caso. O resultado pode ser facilmente encontrado pela regra do trap\303\251zio.
S:=.25/2*(3.91+2*4.11+2*4.32+2*4.54+4.77);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJTR0YoJCIrKytdRlYhIio3I0Yu
O Maple utiliza m\303\251todos similares, embora mais sofisticados, para o c\303\241lculo de aproxima\303\247\303\265es para as integrais definidas. As fun\303\247\303\265es int ou Int apresentam algumas op\303\247\303\265es \303\272teis neste caso. O comando "evalf(Int(f, x=a..b, digits, flag))" retorna um valor num\303\251rico para f , para os limites de integra\303\247\303\243o indicados e o n\303\272mero de d\303\255gitos indicado na op\303\247\303\243o "digits", utilizando tamb\303\251m o m\303\251todo especificado por "flag". A sele\303\247\303\243o destes m\303\251todos requer conhecimento de an\303\241lise num\303\251rica e, portanto, o melhor \303\251 deixar o valor padr\303\243o. O comando evalf for\303\247a a integra\303\247\303\243o num\303\251rica.
Aplica\303\247\303\243o
1. Utilizar o m\303\251todo de Euler para encontrar um solu\303\247\303\243o num\303\251rica e gr\303\241fica para: a. b. c.
2. Utilizar a regra do trap\303\251zio e de Simpson para calcular a integral de sin( x ) entre 0.5 e 0.7. Utilizar intervalos de 0.1 e 0.05.
3. Utilizar o mesmo procedimento para 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 , entre os limites 1.0 e 2.0. Usar os intervalos a sua escolha.
4. Calcular a integral, entre 1 e 2, da curva representada pelos dados a seguir. Utilizar a regra do trap\303\251zio e de Simpson. f(1.0) = 1.442249570 f(1.25= 1.527278686 f(1.50)=1.619805901 f(1.75)=1.717071364 f(2.00)=1.817120593
with(Student[Calculus1]);
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
ApproximateInt(sin(x),0..Pi,method=newtoncotes[2],output=plot,color=black,functionoptions=[color=yellow,thickness=3]);
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