# How to integrate $\int_0^1\int_0^\sqrt{2y-y^2}(1-x^2-y^2)dxdy$

$$\int_0^1\int_0^\sqrt{2y-y^2}(1-x^2-y^2)dxdy$$

I tried to transform it to polar form, but the problem is to find the limits of integration.

$0\le x\le \sqrt{2y-y^2}$ is the right half of the circle $x^2+(y-1)^2=1$ , then $r=2sin(\theta)$. And $0\le y\le 1$ makes the area is the lower-right quarter of the circle.

What are the limits of $r$ and $\theta$ ?

Then the limits of $r$ and $\theta$ are evident from what you wrote.