I have to find the volume of the set $S=: \lbrace{(x,y,z) \in \mathbb{R}^3: z^2(1+x^2+y^2)^2 \le y^2, x^2+y^2 \le 2x \rbrace}$. My attempt: $z \in \left[-\sqrt{\frac{y^2}{1+x^2+y^2}}, \sqrt{\frac{y^2}{1+x^2+y^2}}\right]$
In a turn, $x^2+y^2\le 2x$ is a circle. I can use polar coordinates. I have $ \int\int_S 2\sqrt{\frac{r\sin t}{r^2+1}}drdt$