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Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 3 / 4A / Solution Created 2026-09-23 Updated 2026-10-03

Green's theorem states, for a positively oriented simple boundary

C = \partial R

,

\oint_{C} (P d x + Q d y) = \iint_{R} (Q_{x} - P_{y}) d A .

The defining polynomial becomes

r^{2} (r^{2} - 2 r cos θ - sin^{2} θ),

so the region is the cardioid

0 \leq r \leq 1 + cos θ, - π \leq θ \leq π .

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