Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics V / 1 / a / ii

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- Solution ii

Solution

In polar coordinates, put

x = r cos θ

and

y = r sin θ

. Differentiating along the curve gives

x d y - y d x = r^{2} d θ .

The cardioid has

r \geq 0

and is traversed once counterclockwise as

θ

increases from

0

to

2 π

. Its cusp at

θ = π

causes no difficulty: split the line integral there, or exclude a small arc and take its limit. Thus signed area as a boundary integral gives

∣ D ∣ = \frac{1}{2} \int_{0}^{2 π} (1 + cos θ)^{2} d θ = \frac{1}{2} \int_{0}^{2 π} (\frac{3}{2} + 2 cos θ + \frac{1}{2} cos 2 θ) d θ = \frac{3 π}{2} .

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