I've been stuck on what should be a straight-forward calculation which is making me question whether I actually understand multi-variable calculus. In particular, I always seem to get the wrong answer when I do spherical coordinate integrals. To illustrate, I included (if I followed the image upload directions properly) a picture of how I worked through a calculation of the moment of inertia of a solid, uniform sphere . I consistently get the moment of inertia to be (3/5)MR^2, when every other source says it is (2/5)MR^2.
Surprisingly I can get the right answer if I integrate the sphere by breaking it up into thin disks, but I want to know why the way I'm trying doesn't work yet gets me particularly close to the right answer. I've even checked with Maple to see that I'm solving the integral correctly, so apparently the bounds of the integral don't work.
In case I hit "submit" and the picture didn't work, basically my calculation just involves an integral where the radius goes from 0 to R, the angle around the sphere goes from 0 to 2*pi and the angle from the vertical axis goes from 0 to pi. Every time I get the moment of inertia to be (3/5)MR^2.
I think knowing why this is wrong will also help me understand why many of my other spherical integrals are close but wrong (unless I break it up into thin disks, but I'm too stubborn to solve spherical integrals like that until I know why my way doesn't work.)