Convert the following integral to polar coordinates. You do not need to evaluate. $$\int_{-3}^3 \int_{x}^{\sqrt{9-x^2}} x^2y dy dx$$
My work :
I plotted the limits and I don't understand the bounded region due to $y=x$, but still I got like this which is wrong I know I solved the integral it should be $\frac{-81}{5}$ but the integral in the polar coordinates I obtained is incorrect $$ \int\limits_{\pi/4}^{\pi}\int\limits_{0}^{3}r^4\cos^2 \theta \sin \theta dr d\theta+ \int\limits_{\pi}^{5\pi/4}\int\limits_{-3/\cos \theta}^{-3\sqrt2}r^4\cos^2 \theta \sin \theta dr d\theta$$
Can anyone help me recorrecting it the answer below is not complete, and it is definitely not the double of the answer ???