 Solution

ID: 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

Each radial segment from

0

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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