# Moment of Inertia of a cylinder

## Main Question or Discussion Point

When I am reading a book, I find it is listed that the moment of Inertia of a cylinder is
MR^2/4 + Ml^2/12

It is a cylinder with rotation axis passing through the curve surface and its centre of mass. And its density is constant. With the circile surface raius = R and height = l

Can anybody show me the procedure of the integration? I have tried several times but fail. I just cannot get that answer. It is not homework but I am interested in the process. You know, usually, moment of Inertia is provided in the book, integration is not required. But I am really curious about it. Could anybody please help me?

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tiny-tim
Homework Helper
Hi loup!

Slice the cylinder into discs.

Use the moment of inertia formula for a disc about its diameter, combined with the parallel axis theorem, and integrate.

The problem is I don't know about how to integrate a disc.

And I think the integration of disc actually comes from cylinder. I expected once I finished this cylinder I could do the disc.

tiny-tim
Homework Helper
The problem is I don't know about how to integrate a disc.
Slice it into strips parallel to a diameter, and integrate …

what do you get?

The r requires a cosine and there are more than one variable, what I should do?

I cannot use parallel axis theorem. I think it is too tricky.

tiny-tim