Past exam of the options II course of the University of Oxford / 2012 / H / H2 / d / Solution

Choose the $z$-axis along the cylinder and the origin at its midpoint, so $x^2+y^2\le b^2$ and $-a\le z\le a$. Reflection symmetry puts the centroid at the origin and makes all products of inertia zero. The mass isCylindrical integration gives ${\int }_A\rho x^{2}dV={\int }_A\rho y^{2}dV=m{b}^2/4$, while the uniform height coordinate has ${\int }_A\rho z^{2}dV=m{a}^2/3$. ConsequentlyThe two equal transverse entries reflect the rotational symmetry about the cylinder axis.