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## Homework Statement

calculate the moment of inertia of a sphere of mass M and radius R by integrating over thin shells

## Homework Equations

I

_{shell}=(2/3)mR

^{2}

## The Attempt at a Solution

this is what i have so far

the sphere is decomposed into infinitesimal shells with surface area 4[tex]\pi[/tex]r

^{2}

the mass of each shell is dm=[tex]\rho[/tex](4[tex]\pi[/tex]r

^{2})dr

after expanding rho and canceling terms I get

dm=(3m/r)dr

I=[tex]\int[/tex](2/3)(dm)r

^{2}dr from 0 to R.

I=2mrdr from 0 to R

this gives me I=mr

^{2}

does anyone see what I did wrong?

PS. those "pi" are not supposed to be powers. sorry, i'm not sure how to change them