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Moment of Inertia of a Hollow Cylinder

1. Homework Statement
A hollow cylinder has mass m, an outside radius R2, and an inside radius R1. Use integration to show that the moment of inertia about its axis is given by I = 1/2*m(R2^2 + R1^2)


2. Homework Equations

dm = rho*dV = 2*pi*rho*h*r*dr

3. The Attempt at a Solution

This doesn't really concern the solution of the problem. There's something else that's bugging me. If, ultimately, the solution and the moment of inertia itself in this case doesn't depend on h (because the mass is distributed evenly along h?), why the need to express an infinitesimal element, dm, of the body by using the volume?

We know that the moment of inertia for a solid cylinder is the same as that of a thin circular plate. And so, in finding the moment of inertia of a solid cylinder, I = 1/2*MR^2, one doesn't have to concern oneself with its height.

I guess my question then is, why one cannot express an infinitesimal element of the hollow cylinder by using area instead of volume?
 
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mass is identified in case of cylinder on volume basis and in case of thin plate on area basis.In case of cylinder mass will depend on height of cylinder
 

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