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## Homework Statement

Find the moment of inertia of a spherical shell

## Homework Equations

## The Attempt at a Solution

So I've been mulling over this for about two hours now and haven't figured out where I've made a mistake. Here's what I've done:

I know I can do the integral by doing ##\int ^{r} _{-r} \frac{8}{3}\pi \rho da##, with ##da## a surface element, but I wanted to do it sort of "from scratch" sort of.

I start with ##\int r^2 dm = \int r^2 A(r) dz## and then sub ##R \cos \theta = r## (the distance from ##dm## to ##z##) and ##A(r) = 2 \pi r \rho = 2 \rho\pi R \cos \theta## (circumference of the disk as a function of theta). We then have ##\int ^{R} _{-R} \rho\pi R^3 \cos ^3 \theta dz##. Next we sub in ##dz = \frac{r d \theta}{\cos \theta}## to get ##\int ^{\pi / 2} _{-\pi/2}2 \pi\rho R^4 \cos ^2 \theta d\theta ##. This doesn't come out to the correct expression, clearly.

I know there must be some dumb mistake in there, but I just can't seem to find it. Any help would be appreciated.

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