Engineeing Dynamics - Inertia Dyad of Half Cylinder

[Green Lantern]

I'm getting desperate. The professor has assigned a project and has not clearly explained how to derive the answers. I'm doing the best I can but his TA's and recommended tutors for the class are always incapable of reproducing the answers either. It's a junior level dynamics class, but he's actually turned the class into a machine dynamics class taught in senior or masters level. The work needs to be done in mathematica, but all I need is help with theory since that's the part we don't learn in lecture.

Anyways, here is the question...

Homework Statement

Develop the mass center expression and the inertia matrix and inertia dyadic for a half cylinder, then let the inner diameter approach the outer diameter to develop the same for a thin half shell. Use Mathematica for your work.

Homework Equations

I_i,i = ∫_m(r_j²+r_k²)dm

The Attempt at a Solution

First is the Inertia-matrix of the half cylinder. I'm not sure how to derive all the terms, but I did my best.

{y² + z², -x y, -x z}
{-x y, x² + z², -y z}
{-x z, -y z, x² + y²}

Then I write my position vector:

BrP = x b[1] + y b[2] + z b[3]

Where B is a point in the body frame, P is the endpoint under evaluation, and r is the vector r. b[1,2,3] are unit vectors in the body frame.

Now is where I think I went wrong, if not before. The Volume is:

V = ∫₀^W∫₀^H∫_-L/2^L/2rdxdydz

Then mass is:

m = ∫∫∫ρrdxdydz

So center of mass is:

C = (1/m)∫∫∫ρr(x,y,z)rdxdydz

Then I get lost even more...

S.O.S

 

[SteamKing]

If x,y,z are cartesian coordinates, dV = dxdydz, not r dxdydz