How to Compute the Inertia Tensor of a Cross-Hanger?

[ppedro]

I've done many exercises about inertia tensors of 3D bodies and sticks but now I have this exercise and I got stuck without any idea of how to do the integration to compute the inertia tensor. The statement is this:

"Compute the inertia tensor of a cross-hanger consisting of 3 thin and linear wires, with mass M and length L, glued perpendicularly by their central parts which is placed at (x,y,z)=(0,0,0)."​

As you know, the inertia tensor (or matrix) for an homogeneous body is constructed from

I_{ij}=\int\rho(\delta_{ij}\sum_{k}x_{k}^{2}-x_{i}x_{j})dV

So that, for example,

I_{11}=\int\rho(x_{2}^{2}+x_{3}^{2})dV

If it was a cube for example we could integrate the expression effortlessly with the integral limits in the 3 variables going from -L/2 to L/2. But with this shape I'm not sure how to do it. Can you help me with the integration? Should I fix one variable at 0 and integrate the others?

 

[voko]

The inertia tensor of a body that consists of multiple parts is the sum of the inertia tensors of the parts.