Concept of Moment of Inertia and its limits of integration

[Yam]

Homework Statement

I am trying to work the moment of inertia for

a) rotating rod, axis through the centre of the rod
http://hyperphysics.phy-astr.gsu.edu/hbase/mi2.html#irod3

b) Solid cylinder
http://hyperphysics.phy-astr.gsu.edu/hbase/icyl.html#icyl2
[/B]

Homework Equations

I = R^2 dM

The Attempt at a Solution

I understand all of the steps, except the limits of integration in terms of R or L.

For the rod, the limit of integration is from L/2 to -L/2

Why is the limit of integration for the cylinder from R to 0?
Should it be from the end to end? R to -R?[/B]

 

[Orodruin]

You need to fix your integration limits such that you integrate over the entire body once. In the case of the cylinder, you integrate from zero, but you allow the angular coordinate to vary between zero and 2pi, thus covering the full lap.

 

[Yam]

but you allow the angular coordinate to vary between zero and 2pi

I don't quite understand this part, when did i allow the angular coordinate to vary between zero and 2pi? What is meant by the angular coordinate?

 

[Orodruin]

It is implicit when you multiply by 2pi in $dV = 2\pi L r dr$. The integral is really a volume integral, but the direction along the cylinder axis and the angular direction are trivial since the integrand does not depend on them, thus they are replaced by the size of their integration ranges, $L$ and $2\pi$, respectively.

 

[Yam]

I understand now, thank you for your help! cheers.