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## Homework Statement

Find the inertia tensor for a uniform, thin hollow cone, such as an ice cream cone, of mass M, height h, and base radius R, spinning about its pointed end.

## Homework Equations

[itex]I_{zz} = \sum x^{2}+y^{2}[/itex]

[itex]\rho = \sqrt{x^{2}+y^{2}}[/itex]

## The Attempt at a Solution

I first tried to think of this as a bunch of little rings

[itex]area = 2\pi\rho dz[/itex] as rho is changing as height increases

i defined rho as [itex]\rho=\frac{z R}{h}[/itex]

but how would i set up this integral? My book shows a few examples of doing it with a solid but i don't know how to do it with an area. so far i have

[itex]\int \frac{2\pi R^{3}}{h^{3}} z^{3}dz[/itex]

I know that that the [itex]I_{zz}= \frac{MR^{2}}{2}[/itex] just need help getting started