Question: Consider the solid $V$ of unit density, where $V$ is the region inside the sphere $x^2 + y^2 + z^2 = 4k^2$ and outside the cylinder $x^2 + y^2 = k^2$.
Using cylindrical coordinates, set up and calculate the moment of inertia of $V$ about the $z$-axis. Show all calculations.
Answer: I set up the integral as follows, $$I_z = \int_{0}^{2π}\int_{k}^{2k}\int_{-\sqrt{ 4k^2 -\rho^2}}^{\sqrt{ 4k^2 -\rho^2}}\rho^3dzd\rho d\phi$$
Solving this integral, I obtained,$$ I_z = \frac{44π\sqrt 3 k^5}{5} $$ I'm a bit dubious as to whether this is correct because when I'm hesitant when using the equation for moment of inertia.
Any help would be appreciated.
Thanks