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## Homework Statement

Beginning with Icm = Integral of r^2 dm from r1 to r2, find the moment of inertia of a solid sphere about any tangential axis.

## Homework Equations

Icm = Integral of r^2 dm

## The Attempt at a Solution

I set up the infinitesimally mass of an infinitesimally thin "shell" of the sphere:

dm = 4ρπr2 dr

And then solved for the moment of inertia:

I = ∫r2dm

= ∫r2(4ρπr2 dr)

= 4ρπ∫r4 dr

= (4/5)ρπr5

And solving for ρ we get the following:

ρ = M/((4/3)πr3).

Substituting that into the previously solved equation for I, I get the following:

I = (3/5)Mr3.

What am I doing wrong? I know the formula involves a coefficient of 2/5, not 3/5, but I can't find my problem.

Thank you in advance!