$$\int \int y^2\sin(x^2+y^2)dxdy$$ calculate over D region which is defied as :
$D:{(x,y) x^2+y^2\leq\pi,|x|\leq y}$
(I sketched the region)
we started to study it today and teacher solved in a very peculiar way..
$y=rsin\theta , x=rcos\theta , 0 \leq r \leq \pi , \pi/4 \leq \theta \leq 3\pi/4 $
coverting the integral to polar coordinates we get:
$$\int_{0}^{\sqrt \pi} \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}r^2\sin^2\Theta \sin(r^2)r drd\Theta $$ then he did something which I don't fully understand. He wrote the intgral in the following way : $$\int r^3\sin(r^2)dr \int \sin^2\theta d\theta$$ Is it valid to write the integral like this(dividing the integral into 2 inetgrals regarding the fact that there is multiplication in the original integral)?