Deriving Moment of inertia for a hollow sphere

[kaitlync]

Homework Statement

How do you derive the moment of inertia for a hollow sphere?
I am not ending up with what i need to get which is: (8$$pi$$r^5)/9

Homework Equations

I am not sure if the bounds are correct or if we need to factor something else in.

The Attempt at a Solution

A double integral of the area multiplied by the height.
for our integrals we do from 0 to 4$$pi$$r^2 and the other integral is from 0 to r. The equation in the integral is r^2 + z^2 dzdA. (we get r^2 because x^2+y^2=r^2)

solving that out and get (16$$pi$$r^5)/3.

plugging that in doesn't give the 2/3 we need for the moment of inertia.

 

[ppzmis]

Easier just to point to a link than write it out

http://cnx.org/content/m14292/latest/