= Area and length of a cycloid arch
{title2=$\mathcal A=3\pi a^2,\quad L=8a$}
For the <cycloid> $x=a(t-\sin t)$, $y=a(1-\cos t)$, $a>0$, $0\leq t\leq2\pi$, the area between the arch and its baseline is
$$
\mathcal A=\int y\,dx=a^2\int_0^{2\pi}(1-\cos t)^2dt=3\pi a^2.
$$
Its <arc length> is
$$
L=\int_0^{2\pi}\sqrt{x'(t)^2+y'(t)^2}\,dt
=2a\int_0^{2\pi}\sin(t/2)dt=8a.
$$
The positive sign of $\sin(t/2)$ inside this parameter interval removes the <absolute value> in the speed. Omitting the two endpoint cusps does not change either <integral>.
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