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 Incoming links: Area and perimeter of a cardioid

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / 15 / b / Solution 2026-10-18

Each radial segment from

0

to

1 + cos θ

lies inside the cardioid. Integrating the polar area element gives

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

The three integrals are respectively

2 π

,

0

and

π

. Thus

A = \frac{3 π}{2}

in square coordinate units, agreeing with the scale-one case of the area and perimeter of a cardioid.

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