Calculating Moment of Inertia Tensor for Rod Along x Axis

[w3390]

Homework Statement

A thin rod has mass M and length L. What is the moment of inertia tensor about the center of mass if placed along the x axis.

Homework Equations

I would write the inertia tensor in component notation, but I don't know how to use Latex.

The Attempt at a Solution

I am getting lost in the inertia tensor equation. For example, when I try to find the I_11 component, plugging in I get:

I_11 = int[rho((x_2^2 -x_1^2)+(x_3^2-x_1^2))dV]

I_11 = iiint[rho(-2x_1^2+x_2^2+x_3^2)dx_1dx_2dx_3]

This is where I am stuck. I do not understand what to use as my various bounds or if this equation is even correct.

Any help would be much appreciated.

 

[w3390]

Okay, so I think I figured out my problem. I was doing something wrong with the component form.

Here is what I got for my inertia tensor:

I_11 = 0
I_22 = (1/3)ml^2
I_33 = (1/3)ml^2
I_12 = I_21 = 0
I_13 = I_31 = 0
I_23 = I_32 = 0So it looks something like this:

[..0...0.....0...]
[..0...(1/3)ml^2...0...]
[..0...0...(1/3)ml^2]

Does this look correct?

 

[sgd37]

it should be 1/12 ml^2 you are doing it about the center