Moment of Inertia of a cylinder

[loup]

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?

 

[tiny-tim]

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.

 

[loup]

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

 

[loup]

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]

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?

 

[loup]

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

 

[loup]

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

 

[tiny-tim]

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

uhh? what equation are you using?