C\303\241lculo IILi\303\247\303\243o 26: Comprimento de Arco de Curvas Param\303\251tricasNesta folha de trabalho, utilizaremos o processo de integra\303\247\303\243o para calcular o comprimento de curvas param\303\251tricas planas. Com este mesmo procedimento \303\251 poss\303\255vel encontrar o comprimento de curvas tridimensionais, que n\303\243o ser\303\241 considerado neste estudo. Seja a curva param\303\251trica descrita pelas equa\303\247\303\265es LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYjNiUtRiw2JVEiZkYnRjRGNy1GOzYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y+RkBGQ0ZFRkdGSUZLRk0vRlBRJjAuMGVtRicvRlNGZ24tSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJ0RidGNEY3Rj5GK0Yr, 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, t em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobWZlbmNlZEdGJDYmLUYjNiUtSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJiRidGNEY3Rj4vJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRic=.f := t-> t^2 ; g := t-> t^3 - 3*t ;plot([f(t), g(t), t=-2..2]);O primeiro passo consiste em repartir o intervalo de par\303\242metros [a, b] em n subintervalos iguais, com extremos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSR0XzBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSI9RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlEtRiw2JVEiYUYnRjRGN0Yr, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEkdF8xRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn, ... , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSR0X25GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSI9RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlEtRiw2JVEiYkYnRjRGN0Yr , e tra\303\247ar em seguida a curva linear por partes composta de segmentos de reta que unem os pontos correspondentes na curva original. Pode-se mostrar que isto pode ser feito com o mesmo procedimento pl, utilizado anteriormente para calcular o comprimento de arco de um gr\303\241fico: basta aplicar pl a f e g, separadamente.pl := proc(f,a,b,n) local u,v,k; k := 1 + floor(n*(x-a)/(b-a)); u := a + (k-1)*(b-a)/n; v := a + k*(b-a)/n; unapply(f(u) + (f(v) - f(u))/(v-u)*(x-u), x); end proc:plf := pl(f,-2,2,6):plg := pl(g,-2,2,6):plot({[f(t),g(t),t=-2..2], [plf(t),plg(t),t=-2..2]}, thickness=2);(A sa\303\255da gerada pelas defini\303\247\303\265es de plf e plg foram omitidas, pois n\303\243o s\303\243o de interesse.)O c\303\241lculo do comprimento da curva linear por partes \303\251 bastante simples. Um segmento t\303\255pico une os pontos (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, 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) e (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSR0X2pGJ0Y0RjdGPkYr, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJnRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSR0X2pGJ0Y0RjdGPkYr), de forma que seu comprimento pode ser calculado pela f\303\263rmula da dist\303\242ncia. A adi\303\247\303\243o destes comprimentos --- um para cada subintervalo --- resulta na f\303\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 .O procedimento a seguir calcula esta express\303\243o, dadas as fun\303\247\303\265es f e g, o intervalo de par\303\242metros [a, b] e o n\303\272mero de subintervalos.approxlen := proc(f,g,a,b,n)
local l;
l := (b-a)/n;
sum( sqrt( (f(a+j*l) - f(a+(j-1)*l))^2
+ (g(a+j*l) - g(a+(j-1)*l))^2 ), j=1..n);
end proc:Como sempre, o Maple calcula a resposta de forma exata, incluindo ra\303\255zes quadradas na resposta. Neste caso, \303\251 conveniente exibir uma aproxima\303\247\303\243o decimal.evalf(approxlen(f,g,-2,2,5));evalf(approxlen(f,g,-2,2,10));evalf(approxlen(f,g,-2,2,20));evalf(approxlen(f,g,-2,2,50));evalf(approxlen(f,g,-2,2,100));Observa-se um padr\303\243o familiar: a medida que o n\303\272mero de subintervalos aumenta, os comprimentos das aproxima\303\247\303\265es lineares por partes aproximam-se de um valor fixo, que corresponde ao comprimento da curva original.<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Quest\303\243o 1</Font></Text-field>Utilizar aproxima\303\247\303\265es lineares por partes para encontrar o comprimento da curva acima, com precis\303\243o de 0.01. (Dica: Quais s\303\243o os dois valores de t no ponto de auto-interse\303\247\303\243o?)Solu\303\247\303\243o: A curva cruza o eixo x quando 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 e, novamente, quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkmbXNxcnRHRiQ2Iy1JI21uR0YkNiRRIjNGJ0Y+Ris=, no ponto x = 3. O comprimento da curva entre estes dois valores de t pode ser aproximado utilizando-se approxlen:evalf(approxlen(f,g,-sqrt(3),sqrt(3),5));evalf(approxlen(f,g,-sqrt(3),sqrt(3),10));evalf(approxlen(f,g,-sqrt(3),sqrt(3),50));evalf(approxlen(f,g,-sqrt(3),sqrt(3),100));\303\211 aconselh\303\241vel experimentar um n\303\272mero maior de subintervalos para assegurar que a segunda casa decimal permacer\303\241 est\303\241vel. No entanto, assumiremos que o comprimento da curva \303\251, de fato, 10.73, com uma precis\303\243o de 0.01.<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Quest\303\243o 2</Font></Text-field>Tra\303\247ar a curva polar LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRKCZ0aGV0YTtGJy9GNUZCRj5GKw== . Descrev\303\252-la como uma curva param\303\251trica, e encontrar o comprimento da parte da curva entre LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSgmdGhldGE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiRRIjBGJ0Y3Ris= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSgmdGhldGE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1GIzYlLUkjbW5HRiQ2JFEiMkYnRjctRjs2LVExJkludmlzaWJsZVRpbWVzO0YnRjdGPkZARkJGREZGRkhGSi9GTVEmMC4wZW1GJy9GUEZlbi1GLDYlUSNQaUYnRjRGN0YrRis=.
Solu\303\247\303\243o. Esta curva polar possui as equa\303\247\303\265es 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, 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.f1 := theta-> theta*cos(theta) ;g1 := theta-> theta*sin(theta) ;plot([f1(theta), g1(theta), theta=0..2*Pi]);evalf(approxlen(f1,g1,0,2*Pi,50));evalf(approxlen(f1,g1,0,2*Pi,100));evalf(approxlen(f1,g1,0,2*Pi,500));O comprimento da curva \303\251 aproximadamente igual a 21.25.Dado o poder computacional do Maple, basta utilizar o procedimento num\303\251rico approxlen para obter-se os comprimentos de arco com a precis\303\243o desejada. Frequentemente, um procedimento num\303\251rico deste tipo \303\251 a \303\272nica forma pr\303\241tica para o c\303\241lculo do comprimento de arco, embora tamb\303\251m seja poss\303\255vel utilizar uma integral para o c\303\241lculo do comprimento exato da curva. (Na pr\303\241tica, a integral deve ser calculada numericamente para a maior parte das curvas de interesse.) Com base no Teorema do Valor M\303\251dio, tem-se a 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 .O procedimento a seguir retorna esta integral. Como muitas integrais para o c\303\241lculo do comprimento de arco n\303\243o podem ser resolvidas simbolicamente, arclength retorna a integral n\303\243o avaliada; deve-se utilizar evalf a fim de obter-se uma aproxima\303\247\303\243o decimal para este valor.arclength := proc(f,g,a,b)
Int(sqrt(diff(f(t),t)^2 + diff(g(t),t)^2), t=a..b);
end proc:Para o exemplo em quest\303\243o, verifica-se que a f\303\263rmula da integral encontra-se em acordo com o comprimento de arco obtido pela aproxima\303\247\303\243o linear por partes:f := t-> t^2 ; g := t-> t^3 - 3*t ;evalf(arclength(f,g,-2,2));<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Quest\303\243o 3</Font></Text-field>Verificar se a f\303\263rmula da integral encontra-se em acordo com os resultados obtidos pelas aproxima\303\247\303\265es lineares por partes das quest\303\265es 1 e 2.Solu\303\247\303\243o.evalf(arclength(f,g,-sqrt(3),sqrt(3)));evalf(arclength(f1,g1,0,2*Pi));Estes dois valores est\303\243o em acordo com as aproxima\303\247\303\265es obtidas nas Quest\303\265es 1 e 2, ao menos para as primeiras casas decimais. O uso de mais subintervalos nas Quest\303\265es 1 e 2 poderiam gerar aproxima\303\247\303\265es melhores.