So now that you've seen how to make a disk out of a bunch of hoops, we could instead make a spherical shell out of a bunch of then also. It's kind of like a technique in pottery where you slowly add little rings of clay of different sizes, until you have a beautiful vase! Oh shut up!
So we can do that here too. Pottery and physics meet. On the other hand I was never good much at pottery. You get the size of a ring off by a factor of two and it ends up looking like a moldy lump of clay. The same is true of the math involved in this example. I could go through and do it, but it's a bit tedious. There's a much much more elegant way of calculating the moment of inertia in this example. It requires you to think a lot more, but it requires you to write a lot less.
It uses the symmetry of sphere. Let's write things out in terms of discrete masses because it's easier to understand
If we rotate about the z axis, then 
is the distance between the
point and the z axis, so 
. So
We could instead compute what I'll call 
or
Because of the symmetry of a sphere we can replace x by y and nothing should change so
I could also calcluate
That should also be the same as , again because of symmetry. There
is nothing special about the choice of axis. We could call x y, y z, and z
x, and we'd get the same answers.
Now lets calculate 
. That's
But since we have a sphere, we know that 
.
So we can pull that out of the sum and then we just have a sum over the
's which just equals M. So 
. But 
.
So
