Cycloid - Area

Area

One arch of a cycloid generated by a circle of radius r can be parameterized by

$\begin{align} x &= r(t - \sin t) \\ y &= r(1 - \cos t) \end{align}$

with

Since

we find the area under the arch to be

$\begin{align} A &= \int_{t=0}^{t=2 \pi} y \, dx = \int_{t=0}^{t=2 \pi} r^2(1 - \cos t)^2 dt \\ &= \left. r^2 \left(\frac{3}{2}t - 2\sin t + \frac{1}{2} \cos t \sin t\right) \right|_{t=0}^{t=2\pi} \\ &= 3 \pi r^2 \end{align}$

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