Suppose a rigid body has mirror symmetry along the $z$-Axis, i.e. $\rho(x,y,z)=\rho(-x,-y,z)$ where $\rho$ is the density of the body.
How can I show from this that the center of mass lies on the $z$-Axis and that those non-diagonal entries of the inertia tensor corresponding to $z$ vanish?
Both statements are very intuitive, but I would like to prove it formally.
I thought that maybe cylindrical coordinates would help, but those don't get me anywhere either.
Any hint or advice is very much appreciated!