Why Does Extracting Density Affect My Moment of Inertia Calculation?

[kuahji]

I'm attempting to derive the moment of inertia for a cylindrical object.

I know that I=\int r^2 dm

which equals =\int r^2 p dV

My question begins here, the derivations I seen pull p out of the integral, which makes sense to do, because in this case it's a constant. p=M/(\pir^2L). So if I don't pull p out before integrating I get I=Mr^2, if I do pull it out, I get I=M/2r^2. I know the answer should be I=M/2r^2 because I have a solid cylindrical object. So why am I getting a different result when I leave p in, & a different result when I pull p out or am I just making a silly math error?

Below is my work when I leave p inside the integral

I=\int r^2*p*(2\pi*r)dr
=2M\int r dr (replacing p with M/(\pir^2L) before integrating)
=Mr^2

 

[ApexOfDE]

I believe that density is constant for each material.

p/s: you replace m = DV, then you replace D = M/V... I don't get it :(

 

[kuahji]

I replaced dm with pdV & then p with M/V.