# MoI of a Sphere using Spherical Coordinates

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## Homework Statement

Calculate the moment of inertia of a uniformly distributed sphere about an axis through its center.

## Homework Equations

I know that
$$I= \frac{2}{5} M R^{2},$$
where ##M## is the mass and ##R## is the radius of the sphere. However, for some reason,
when I do this integration in spherical coordinates, I do not get this result.

## The Attempt at a Solution

The density is
$$\rho= \frac{M}{V}= \frac{3M}{4 \pi R^{3}}.$$
Then the moment of inertia is
$$I= \int r^{2} \,dm= \rho \iiint_{V}r^{2} \,dV = \rho \int_{0}^{2 \pi} \int_{0}^{ \pi} \int_{0}^{R} r^{4} \sin(\theta) \, dr \, d\theta \, d\phi =4 \pi \rho \frac{R^{5}}{5}$$
$$=4 \pi \frac{3M}{4 \pi R^{3}} \frac{R^{5}}{5}= \frac{3}{5} MR^{2}.$$
Where is my error?

I know, I know: all the derivations of this result I can find split the sphere up into
disks. I understand those derivations: I want to know where this one is wrong.

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