 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ia/paper-3/3a/ii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Write the upper cap as

z = 1 - x^{2} - y^{2}

. Its area factor is

(1 - r^{2})^{- 1/2}

, and the projected disc has radius

a

. Therefore

area (S) = 2 π \int_{0}^{a} \frac{r}{1 - r ^{2}} d r = 2 π (1 - 1 - a) .

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