 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2005/paper-1/8/b/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

x = r cos θ

y = r sin θ

, and the Jacobian determinant contributes

r

. The integrand and area element together become

r^{2} e^{- r^{2}} sin θ cos^{2} θ d r d θ

. Hence

I = (\int_{0}^{π /2} sin θ cos^{2} θ d θ) (\int_{0}^{\infty} r^{2} e^{- r^{2}} d r) = \frac{1}{3} \frac{π}{4} = \frac{π}{12} .

The radial integral follows by integration by parts from the Gaussian integral. The isolated undefined ratio at the origin does not affect this convergent double integral.

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