I want to know how to calculate the volume bound by a surface $f(x,y)$ in cylindrical coordinates. As an example, for a sphere of radius $r$, I would use $f(x,y) = \sqrt{r^2 - x^2 - y^2}$, and $V = \int^r_{-r}\int^r_{-r} f(x,y)dxdy$ in Cartesian coordinates.
To do the same in cylindrical coordinates, I do $x\rightarrow x$, $y\rightarrow \rho\sin\theta$, $z\rightarrow \rho\cos\theta$. To calculate the volume, I can then do $V = \int_0^{2\pi}\int_{-r}^r \rho^2 (x,\theta)dxd\theta$, where $\rho(x,\theta)$ is the cylindrical radius corresponding to the surface $f(x,y)$: $x^2 + \rho^2 = r^2 \Leftrightarrow \rho^2 (x,\theta) = r^2 - x^2$. One $\rho$ comes from the $f(x,y)$, and the other from going to cylindrical coordinates. When I calculate this integral, however, the outcome is $V = \frac{8}{3}\pi r^3$, which is off by a factor 2. What am I doing wrong here?
This is because the limits $0<\theta <2\pi$ cover two times the volume: one for the values that gives $r\ge 0$ and the other for the values that gives $r<0$