In case you're interested, this is what we did in class today in finding a moment of inertia of a cube with respect to any
axis passing through its center of mass, provided that the mass density,
, is constant.
be the position vector of any infinitesimal mass form the centre of the cube,
be a unit vector pointing along an arbitrary axis, and
be the distance from the axis to the infinitesimal mass.
It is clear that the integrals of the cross terms vanish.
Notice that this is independent of an axis chosen.