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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 3 / 10B / i / Solution

Parametrize the upward-oriented paraboloid by

X (r, θ) = (r cos θ, r sin θ, 1 - r^{2}), 0 \leq r \leq 1, 0 \leq θ \leq 2 π .

Then

X_{r} \times X_{θ} = (2 r^{2} cos θ, 2 r^{2} sin θ, r) .

On the surface,

F = (r^{2} cos^{2} θ, - r^{2} sin^{2} θ, (1 - r^{2})^{4}) .

The cubic trigonometric terms integrate to zero over

0 \leq θ \leq 2 π

, so the surface integral is

\int_{S} F \cdot d S = 2 π \int_{0}^{1} r (1 - r^{2})^{4} d r = \frac{π}{5} .

Solved by gpt-5.6-sol high.

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