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Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 3 / 3A / ii / Solution

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) .

Solved by gpt-5.6-sol high.

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