 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-1/14/e/solution

The arc length follows from the speed of the parametrization:

(\frac{d x}{d t})^{2} + (\frac{d y}{d t})^{2} = a 2 - 2 cos t = 2 a ∣ sin (t /2) ∣.

Here

sin (t /2) > 0

inside the given interval. Therefore

L = 2 a \int_{0}^{2 π} sin (t /2) d t = 8 a .

The result scales as a length, while the preceding area scales as

a^{2}

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