Q:

Tough Moment of Inertia Problem About a Super Thin Spherical Shell Using Spherical Coordinates

I need to compute the moment of inertia of a spherical shell with radius $R$, constant density $\rho$, and total mass $M$ throughout some (any) axis through the origin. 
The question specifies that this should be a double integral, since it's a super thin shell. So that means I'd need to use $dA = R^2\sin{\phi}d{\phi}d{\theta}$. 
From here, though, I'm stuck, and even unsure where to start. Any help would be much appreciated. 

A:

The mass of the $dA$ area is $dm=\rho dA=\frac{M}{4\pi R^2}dA$. I assumed here that $\rho$ is a surface density. You can calculate the moment of inertia with respect to any axis, they are all equal. Then for simplicity, use the axis where $\phi=0$. The distance from this axis is $r=R\sin\phi$, so $$I=\iint r^2dm=\int_0^{2\pi} d\theta\int_0^\pi\frac{M}{4\pi R^2}R^4\sin^3\phi$$

