Question

Calcule a integral dupla com as seguintes informações:

Answer 1

Queremos calcular

$\displaystyle\sf{\int_{1}^{2}\int_{0}^{\pi}y\,sen(xy)~dx~dy}$

Recorramos a uma substituição simples para avaliar a integral mais interna:

$\sf{t=xy~~\dfrac{dt}{dx}=y\implies dt=x~dy}\\\sf{se~x=0\implies t=0}\\\sf{se~x=\pi\implies t=\pi y}$

Assim a integral dupla equivalente será:

$\displaystyle{\int_{1}^{2}\int_{0}^{\pi y}sen(t)~dt~dy}\\\displaystyle\sf{\int_{1}^{2}-cos(t)\Bigg|_{0}^{\pi y}~dy}$

$\displaystyle\sf{-\int_{1}^{2}\left[cos(\pi y)-cos(0)\right]~dy} \\\displaystyle\sf{-\int_{1}^{2}cos(\pi y)~dy+\int_{1}^{2}dy}$

Lembrando que

$\displaystyle\sf{\int cos(at)dt=\dfrac{1}{a}sen(at)+k}$

Temos:

$\sf{-\dfrac{1}{\pi}\left[\underbrace{sen(2\pi)}_{0}-\underbrace{sen(\pi)}_{0}\right]+y\Bigg|_{1}^{2}}\\\sf{2-1=1}$

Portanto

$\displaystyle\sf{\int_{1}^{2}\int_{0}^{\pi}y\,sen(xy)~dx~dy=1~\checkmark}$