Calculating the moment of inertia of Cone

[EEristavi]

Homework Statement

Calculate the moment of inertia of a uniform solid cone relative to its symmetry axis, if the
mass of the cone is equal to m and the radius of its base to R

Relevant Equations

I = m r^2

I'm Summing the Inertia of "donuts" with width dr and radius - r.
I'm also "flattering" the cone into 2D and considering that each donut has different mass - because of the different height - h

so:
dm = 3 m h / (pi R² H) dr

I = ∫ dm r² = 3 m h / (pi R² H) r² dr

from triangle similarities
H/R = h/(R-r) => h = H - H/R r

afterwards, I'm calculating integral. However, I'm getting wrong answer.

My question:
Is Integral I've written above correct?

 

[kuruman]

Can you show more of your work? Have you tried adding moments of inertia of stacked disks, the radius of which increases linearly with height?

 

[EEristavi]

Maybe This will help.

It's very hard to type in here, sorry...

 

[kuruman]

It's very hard to type in here, sorry...

You should try to learn how to use LaTeX for writing equations. Click on the link "LaTeX Guide" above the "Attach files" link.

Your $dm$ is incorrect because it has the wrong dimensions. It might be easier to set it up if you used a density $\rho=\frac{3m}{\pi R^2 H}$ and replace it at the very end after you integrate.

 

[EEristavi]

found my error.
I was just worried that my idea of writing integral was incorrect.

Thank you also about LaTeX - It will help me a lot :)

Thanks