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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 here. 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.)

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I guess you may have a wrong definition of the moment of inertia. You should integrate the square of the distance from the axis, not the square of the distance from the origin.

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  • $\begingroup$ Yeah, that was it and that makes complete sense. I was just given an out of context integral and this is naturally how I would solve that integral. How do I get back all the time I wasted? $\endgroup$ – Ryan Franz Oct 30 '15 at 5:19
  • $\begingroup$ @RyanFranz: At least you may have learned some lessons:-). For example, in case of a problem, check everything, from A to Z. I heard this story some time ago (it does not matter if it's true or not): some Russians were trying to fix a TV, but, although they were quite experienced in electronics, they could not get any image on the screen, no matter what they tried. It turned out eventually that the TV tower in Moscow was on fire at that moment (en.wikipedia.org/wiki/Ostankino_Tower#Incidents_and_accidents) $\endgroup$ – akhmeteli Oct 30 '15 at 5:34

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