Calcule as integrais dadas: $\int_{0}^{8} \int_{\sqrt[3]{y}}^{2} \operatorname{sen}\left(x^{4}\right) d x d y$.

Passo 1

Primeiro, vamos entender o que a questão está pedindo. Precisamos calcular a integral dupla . Isso envolve o estudo de integrais múltiplas, especificamente integrais duplas, e vamos precisar saber como trocar a ordem de integração e usar substituições adequadas para resolver.

Passo 2

Observando os limites de integração, vemos que para cada valor de entre 0 e 8, varia de até 2. Trabalhar diretamente com esses limites pode ser complicado por causa da função . Então, uma boa estratégia é trocar a ordem de integração para simplificar o cálculo.

Passo 3

Para trocar a ordem de integração, precisamos reescrever os limites. Agora, vai de 0 a 2, e para cada , varia de 0 até . Assim, a integral fica:

Passo 4

Agora, vamos integrar em relação a primeiro. Como não depende de , a integral interna é simples:

Passo 5

Substituindo de volta na integral externa, temos:
Agora, precisamos resolver essa integral.

Passo 6

Para integrar , vamos usar uma substituição. Seja , então:

Passo 7

Substituindo na integral, ficamos com:

Passo 8

Integrando , temos:

Passo 9

Sabemos que , então:

Resposta

A integral dupla é igual a .

Exercícios relacionados

Utilize somas de Riemann com para estimar a integral da função , onde tomando como ponto amostral o ponto médio de cada subretângulo. Calcule também o valor médio da função.
Calcule a integral dupla , em que é a região do plano cartesiano limitada pela reta e pela parábola . Esboce a região.
Calcule as integrais dadas:
<html><head></head><body><div>Utilize integrais duplas para calcular a área:<br>
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