C\303\241lculo IILi\303\247\303\243o 3: Aplica\303\247\303\265es da Integra\303\247\303\243o 1: Trabalho Quando uma for\303\247a move um objeto, diz-se que esta for\303\247a realiza trabalho. Se a for\303\247a F \303\251 constante, ent\303\243o o trabalho realizado \303\251 dado pela equa\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJXRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRI0ZkRidGNEY3Rj5GK0Y+, onde d \303\251 a dist\303\242ncia percorrida. O que acontece quando a for\303\247a n\303\243o \303\251 constante?Suponhamos que uma for\303\247a F(x) mova um objeto de a para b ao longo do eixo x. Embora F dependa de x, pode-se dividir o intervalo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJiRidGNEY3Rj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y+ em pequenos subintervalos e supor que F \303\251 quase constante em cada subintervalo. Vejamos porque a express\303\243o 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 \303\251 um aproxima\303\247\303\243o do trabalho total realizado na movimenta\303\247\303\243o do objeto de a para b. Quando o intervalo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJiRidGNEY3Rj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y+ \303\251 dividido em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= subintervalos iguais, cada subintervalo possui comprimento 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. Quando n \303\251 muito grande, cada subintervalo \303\251 muito pequeno, de forma que a for\303\247a no LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-\303\251simo subintervalo pode ser aproximada por seu valor em qualquer ponto do subintervalo. Escolhemos a extremidade direita do subintervalo como sendo o ponto 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. A for\303\247a no k-\303\251simo subintervalo \303\251, portanto, aproximadamente igual a 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, e o trabalho realizado ao se cruzar este subintervalo, utilizando a f\303\263rmula para a for\303\247a constante, \303\251 aproximadamente 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. O trabalho total realizado no movimento de a para b \303\251 calculado pela adi\303\247\303\243o dos n termos, um para cada subintervalo, resultando assim na f\303\263rmula no enunciado da quest\303\243o. Pode-se agora escrever uma fun\303\247\303\243o, worksum, que recebe uma fun\303\247\303\243o F, um intervalo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJiRidGNEY3Rj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y+ e um n\303\272mero n, e retorna uma aproxima\303\247\303\243o do subintervalo n para o trabalho relizado pela for\303\247a F devido ao deslocamento de um objeto de a para b (esta fun\303\247\303\243o traduz a f\303\263rmula dada na Quest\303\243o 1 em sintaxe do Maple). worksum := (F,a,b,n)-> sum(F(a + k*(b-a)/n)*(b-a)/n, k=1..n) ;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSh3b3Jrc3VtR0YoZio2JkkiRkdGKEkiYUdGKEkiYkdGKEkibkdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoLUkkc3VtR0YlNiQqKC05JDYjLCY5JSIiIiooSSJrR0YoRkAsJjkmRkBGPyEiIkZAOSdGRUZARkBGQ0ZARkZGRS9GQjtGQEZGRihGKEYoNyNGLg==

A for\303\247a exercida em um objeto de massa m na superf\303\255cie da Terra \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JS1JI21vR0YkNi1RKiZ1bWludXMwO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZJLUYsNiVRI21nRicvJSdpdGFsaWNHUSV0cnVlRicvRjZRJ2l0YWxpY0YnRjVGK0Y1, onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJnRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFElOS44MkYnRj5GPkYrRj4= m/LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1GLDYlUSJzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiMkYnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGQUYrRkE= \303\251 a 'acelera\303\247\303\243o da gravidade'. Obviamente, este for\303\247a diminui a medida que objeto distancia-se da Terra. Na verdade, a lei de for\303\247a correta \303\251 dada pela Lei da Gravita\303\247\303\243o de Newton: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 .Neste caso, F \303\251 a for\303\247a exercida em um objeto, m \303\251 sua massa, M \303\251 a massa da Terra, r \303\251 a dist\303\242ncia do objeto ao centro da Terra, e G \303\251 uma constante universal. Os valores de M e G s\303\243o conhecidos, muito embora n\303\243o sejam necess\303\241rios, como demonstra o argumento a seguir.(a) O quil\303\264metro foi originalmente definido como sendo 1/10000 da dist\303\242ncia do Polo Norte ao Equador, ao longo do meridiano que passa por Paris. Tem-se portanto que a circunfer\303\252ncia da Terra \303\251 quase exatamente igual a 40000 quil\303\264metros. Encontrar o raio da Terra, atribuindo-lhe \303\240 vari\303\241vel R. R := evalf(40000/(2*Pi));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJSR0YoJCIrQXg+bWohIic3I0Yu

(b) Na superf\303\255cie da Terra, r = R. Considerando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRIlJGJ0Y0RjdGPkYrRj4= na Lei da Gravita\303\247\303\243o de Newton, obt\303\251m-se uma express\303\243o para a for\303\247a gravitacional na superf\303\255cie da Terra. Igualar esta express\303\243o a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JS1JI21vR0YkNi1RKiZ1bWludXMwO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZJLUYsNiVRI21nRicvJSdpdGFsaWNHUSV0cnVlRicvRjZRJ2l0YWxpY0YnRjVGK0Y1 e, em seguida, encontrar o valor do produto GM. (Aten\303\247\303\243o \303\240s unidades: g \303\251 expresso em termos de metros/LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1GLDYlUShzZWd1bmRvRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiMkYnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGQUYrRkE=, enquanto as outras dist\303\242ncias est\303\243o em quil\303\264metros.)Solu\303\247\303\243o.\303\211 aconselh\303\241vel iniciar com alguns c\303\241lculos manuais. Equacionando as duas express\303\265es para a for\303\247a na superf\303\255cie da Terra, obt\303\251m-se 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. Cancelando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JS1JI21vR0YkNi1RKiZ1bWludXMwO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZJLUYsNiVRIm1GJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU= e resolvendo a equa\303\247\303\243o para o produto GM, obt\303\251m-se 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. Utilizaremos o valor de R calculado acima e o valor de g dado na quest\303\243o mas, para tornar as unidades consistentes, expressaremos g em km/LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1GLDYlUSJzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiMkYnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGQUYrRkE=: g := 0.00982;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJnR0YoJCIkIykqISImNyNGLg==

GM := R^2 * g;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHTUdGKCQiKyM0JyopelIhIiU3I0Yu

(c) A uma dist\303\242ncia de 42377 quil\303\264metros acima do centro da Terra, um sat\303\251lite gira em uma \303\263rbita geoestacion\303\241ria: este leva exatamente 24 horas para realizar uma volta completa em torno da Terra e, portanto, encontra-se sempre diretamente acima do mesmo ponto na Terra. Sat\303\251lites de comunica\303\247\303\265es, por exemplo, devem realizar \303\263rbitas geoestacion\303\241rias. Utilizar a fun\303\247\303\243o do Maple da quest\303\243o 2 e calcular aproxima\303\247\303\265es para a quantidade de trabalho que deve ser realizada a fim de que se possa lan\303\247ar um sat\303\251lite de 250 quilogramas, a partir da superf\303\255cie da Terra, em uma \303\263rbita geoestacion\303\241ria. Utilizar aproxima\303\247\303\265es com 100 e 1000 subintervalos. A fun\303\247\303\243o deve retornar uma valor negativo, embora esta realize uma quantidade de trabalho positiva para o lan\303\247amento do sat\303\251lite em \303\263rbita. Explique este fato.Solu\303\247\303\243o. \303\211 preciso lan\303\247ar o sat\303\251lite a partir da superf\303\255cie da Terra em uma \303\263rbita geoestacion\303\241ria. Como as dist\303\242ncias s\303\243o medidas a partir do centro da Terra, deve-se lan\303\247ar o sat\303\251lite a uma dist\303\242ncia de R a 42377 quil\303\264metros. Deseja-se saber qual o trabalho realizado sobre a for\303\247a da gravidade pela movimenta\303\247\303\243o entre estas dist\303\242ncias. A fun\303\247\303\243o worksum apresenta (aproxima\303\247\303\265es para) o trabalho realizado pela for\303\247a. Primeiramente, deve-se indicar qual a for\303\247a: f := r->-GM*250/r^2 ;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJmR0YoZio2I0kickdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoLCQqJkkjR01HRigiIiI5JCEiIyEkXSNGKEYoRig3I0Yu

worksum(f,R,42377,100);

JCErS1x0JkciISIm

worksum(f,R,42377,1000);

JCErTSMpekI4ISIm

Devemos lembrar que este \303\251 o trabalho realizado pela for\303\247a, ou seja, o valor negativo para o trabalho no lan\303\247amento do sat\303\251lite em \303\263rbita. (A for\303\247a gravitacional realizar\303\241 uma quantidade de trabalho positiva quando o sat\303\251lite voltar a Terra.) Deve-se considerar apenas 3 casas decimais nas respostas, uma vez que o valor de g foi dado com esta precis\303\243o. Digamos, ent\303\243o, que com 1000 subintervalos pode-se estimar a quantidade de trabalho necess\303\241ria em 13200 Newton-quil\303\264metros, ou (em unidades mais usuais) 13200000 Newton-metros.

Nota-se que a express\303\243o 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 \303\251 uma aproxima\303\247\303\243o para o trabalho realizado. Em medidas f\303\255sicas, poder-se-ia esperar que esta aproxima\303\247\303\243o fosse mais e mais precisa com o aumento de n. Por outro lado, matematicamente, reconhecemos que esta aproxima\303\247\303\243o \303\251 uma soma de Riemann para a integral 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, e sabemos que somas de Riemann convergem para esta integral. Portanto, \303\251 razo\303\241vel concluir que a integral fornece a quantidade exata do trabalho realizado pela for\303\247a.

Utilizar uma integral apropriada para o c\303\241lculo da quantidade exata de trabalho necess\303\241ria para o lan\303\247amento de um sat\303\251lite. Comparar com aproxima\303\247\303\265es para a soma de Riemann. (Por exemplo, poder-se-ia perguntar: Qu\303\243o precisas s\303\243o as aproxima\303\247\303\265es? Quantos subintervalos s\303\243o necess\303\241rios para uma aproxima\303\247\303\243o com erro m\303\241ximo de 1%?)Solu\303\247\303\243o. (A fun\303\247\303\243o f \303\251 definida como na Quest\303\243o 3.) exact_work := int(f, R..42377);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SStleGFjdF93b3JrR0YoJCErJFw1Ikc4ISImNyNGLg==

Como na Quest\303\243o 1, este \303\251 o trabalho realizado pela for\303\247a, que \303\251 justamente o negativo do valor desejado. Com uma precis\303\243o de 3 casas decimais, o trabalho necess\303\241rio para o lan\303\247amento do sat\303\251lite \303\251 de 13300000 Newton-metros.As aproxima\303\247\303\265es na Quest\303\243o 1 s\303\243o bastante razo\303\241veis. Em particular, um erro de 1% significa (em Newton-quil\303\264metros) um erro de n\303\243o mais que 132 N-km, ou uma aproxima\303\247\303\243o entre exact_work - 132; exact_work + 132;

JCErJFw1OE0iISIm

JCErJFw1XEoiISIm

A aproxima\303\247\303\243o com 1000 subintervalos certamente satisfaz esta exig\303\252ncia, o que n\303\243o acontece com a aproxima\303\247\303\243o de 100 subintervalos. Quantos subintervalos s\303\243o necess\303\241rios? (O m\303\251todo de tentativa e erro \303\251 usado para obter-se o n\303\272mero de subintervalos nos comandos a seguir.) worksum(f,R,42377,350);

JCErQVEkZUoiISIm

worksum(f,R,42377,300);

JCErIiozIVFKIiEiJg==

Algo em torno de 300 a 350 intervalos s\303\243o suficientes --- pode-se fazer mais tentativas para obter-se maior precis\303\243o..Pergunta: Nesta quest\303\243o, facilmente verifica-se a precis\303\243o das aproxima\303\247\303\265es, j\303\241 que estas podem ser comparadas com a resposta exata. Obviamente, isto torna a compara\303\247\303\243o desnecess\303\241ria: quando se tem a resposta exata, n\303\243o \303\251 preciso preocupar-se com aproxima\303\247\303\265es. As somas de Riemann s\303\243o de maior utilidade quando n\303\243o se tem a resposta exata. Nestas situa\303\247\303\265es, como saber se as aproxima\303\247\303\265es s\303\243o suficientemente precisas?