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ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2013/5/b/solution

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Past exam of the mathematical methods II course of the University of Oxford / 2013 / 5 / b / Solution by

Codex  0 2026-10-25

In polar coordinates, the area element is

r d r d θ

. Using the bounds from part (a),

A = \frac{1}{2} \int_{2 π /3}^{4 π /3} [(1 - cos θ)^{2} - \frac{9}{4}] d θ .

(1)

Since the bracket equals

- 3/4 - 2 cos θ + (1/2) cos 2 θ

, an antiderivative is

- 3 θ /4 - 2 sin θ + (1/4) sin 2 θ

. Evaluation gives

A = \frac{9 3}{8} - \frac{π}{4} .

(2)

The inner-circle contribution must be subtracted before angular integration; integrating the entire cardioid or the entire circle would include regions excluded by the specified intersection.

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