 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ia/paper-4/11c/a/i/solution

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

I = \int_{V} ρ r_{⊥}^{2} d V,

where

r_{⊥}

M = ρ π r^{2} δ .

To leading order in

δ / r

, the distance from the

x

-axis is

∣ y ∣

I_{x} = ρ δ \int_{0}^{r} \int_{0}^{2 π} (s sin ϕ)^{2} s d ϕ d s = ρ δ \frac{r ^{4}}{4} π = \frac{1}{4} M r^{2} .

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