Moment of inertia of a spherical segment

Am I going about this the right way?

There is a sphere of radius 3 and a region that lies between the planes $\displaystyle z=1$ and $\displaystyle z=2$ and has a density of $\displaystyle cz$. We are asked to work in cylindrical coordinates.

Let $\displaystyle \rho=\sqrt{9-z^2}$, is the following the right formula?

$\displaystyle I=\displaystyle{\int_R}cz\rho^2\,dV$ where $\displaystyle dV=\rho\,d\theta\,d\rho\,dz$

Would this be the integral?

$\displaystyle I=2\pi c z\displaystyle{\int^2_1}\displaystyle{\int^{\sqrt{ 9-z^2}}_0}\rho^3\,d\rho\,dz$

Thanks