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 Incoming links: Polar cap between a cardioid and a circle

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / 15 / a / ii / Solution 2026-10-18

The polar area element is

r d r d ϕ

. Integrating first in radius and using symmetry gives

A = \frac{1}{2} \int_{- π /3}^{π /3} [(1 + cos ϕ)^{2} - \frac{9}{4}] d ϕ = \int_{0}^{π /3} [- \frac{3}{4} + 2 cos ϕ + \frac{1}{2} cos 2 ϕ] d ϕ = [- \frac{3 ϕ}{4} + 2 sin ϕ + \frac{1}{4} sin 2 ϕ]_{0}^{π /3} = \frac{9 3}{8} - \frac{π}{4} .

This is the specialization

a = 1

,

c = 3/2

of the polar cap between a cardioid and a circle area formula.

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