Now we have the moment of inertia of a spherical shell, we can sum up all these shells to get what it is for solid sphere. This is a lot like the example of the disk.
So what's the mass dm of a shell of inner radius r and outer radius
r+dr? Call the density 
. Then
. What's the volume dV? It's the surface area of a sphere
of radius r times dr. The surface area of a sphere is 
so
And from the last example, that 
. So the moment of inertia is
Let's write in terms of the M and R. The volume a sphere
is 
, so
Plugging that in to the formula for I
or
