 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ia/paper-4/3c/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

For a lamina in the

x y

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

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

For the uniform ellipse, put

x = a u

y = b v

, where

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

\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}) .

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