Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2020 / Paper 1 / 12 / a / ii

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- Solution ii

Solution

The cardioid has a nonnegative radial boundary, so each ray from the origin contributes

0 \leq r \leq a (1 + cos θ)

. Integrating the polar coordinates area element yields

A = \int_{0}^{2 π} \int_{0}^{a (1 + c o s θ)} r d r d θ = \frac{a ^{2}}{2} \int_{0}^{2 π} (1 + 2 cos θ + cos^{2} θ) d θ .

The three angular integrals are

2 π

, zero and

π

, respectively, using the double-angle formula for the last. Thus

A = \frac{3 π a ^{2}}{2} .

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