Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 4 / 3C / b / Solution

For a lamina in the $xy$-plane,so the perpendicular axis theorem is immediately

For the uniform ellipse, put $x=au$, $y=bv$, where $u^2+v^2\le 1$. Symmetry of the unit disk givesand the same for $v^2$. Hence