Double integral Cartesian to polar coordinate

Question

$\int_{-2}^2\int_{-\sqrt{4-y^2}}^{\sqrt{4-y^2}} dxdy$. I need help to solve this in polar coordinates. I am confused about polar coordinates. What will be the limits? Thanks in advance for your kind help.

Answer 1

The constraints $-2\le y\le2,\,-\sqrt{4-y^2}\le x\le\sqrt{4-y^2}$ define the circle (closed disk, if we're pedantic) of centre $O$ and radius $2$, so the integral is that circle's area. That saves you manually computing $\int_0^{2\pi}\int_0^2rdrd\theta$.