Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 4 / 9B / a / Solution

Take the sheet to lie in the $x_1{x}_2$ plane, with lengths $a,b$ in those directions. Its surface density is $M/(ab)$. Reflection symmetry about the coordinate axes makes the off-diagonal entries of the inertia tensor zero. Its principal moments of inertia areand similarly $I_2=M{a}^2/12$. Since $I_3$ integrates $x_1^2+x_2^2$, the perpendicular axis theorem gives