 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/d/solution

Because

d x / d t > 0

along the arch, its area above the baseline is the definite integral

\int y d x

. Changing to the parameter gives

A = a^{2} \int_{0}^{2 π} (1 - cos t)^{2} d t = a^{2} [\int_{0}^{2 π} 1 d t - 2 \int_{0}^{2 π} cos t d t + \int_{0}^{2 π} cos^{2} t d t] = a^{2} (2 π + π) = 3 π a^{2} .

Adding or removing the two endpoint points does not change the area. This is the area formula in area and length of a cycloid arch.

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