How Is the Moment of Inertia Tensor Calculated for a Cuboid?

[Logarythmic]

Homework Statement

Compute the moment of inertia tensor I with respect to the origin for a cuboid of constant mass density whose edges (of lengths a, b, c) are along the x,y,z-axes, with one corner at the origin.

The Attempt at a Solution

I get

$$I = M \left( <br /> \begin{array}{ccc}<br /> \frac{1}{3} (b^2+c^2) & -\frac{ab}{4} & -\frac{ac}{4}\\<br /> -\frac{ab}{4} & \frac{1}{3} (a^2 + c^2) & -\frac{bc}{4}\\<br /> -\frac{ac}{4} & -\frac{bc}{4} & \frac{1}{3} (a^2 + b^2)<br /> \end{array}<br /> \right)$$

Can this be right?

 

[radou]

I think it may be right, since the matrix is supposed to be symmetric.

 

[Logarythmic]

Yeah, but it can be symmetric in many ways. ;)

Can someone please explain the equality

$$\int_V \rho(\vec{r}) (r^2 \delta_{jk} - x_jx_k) dV = \int_V \rho(x,y,z)<br /> \left(<br /> \begin{array}{ccc}<br /> y^2+z^2 & -xy & -xz\\<br /> -xy & z^2+x^2 & -yz\\<br /> -xz & -yz & x^2+y^2<br /> \end{array}<br /> \right)dxdydz$$

for me? I think this is the most important step in my understanding for this problem.