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

For a lamina in the

x y

-plane,

I_{x} = \int y^{2} d m, I_{y} = \int x^{2} d m, I_{z} = \int (x^{2} + y^{2}) d m,

so the perpendicular axis theorem is immediately

I_{z} = I_{x} + I_{y} .

For the uniform ellipse, put

x = a u

,

y = b v

, where

u^{2} + v^{2} \leq 1

. Symmetry of the unit disk gives

\frac{1}{π} \int_{u^{2} + v^{2} \leq 1} u^{2} d u d v = \frac{1}{4},

and the same for

v^{2}

. Hence

I_{x} = \frac{M b ^{2}}{4}, I_{y} = \frac{M a ^{2}}{4}, I_{z} = \frac{M}{4} (a^{2} + b^{2}) .

Solved by gpt-5.6-sol high.

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