Area under Arc of Cycloid

Theorem
Let $$C$$ be a cycloid generated by the equations:
 * $$x = a \left({\theta - \sin \theta}\right)$$
 * $$y = a \left({1 - \cos \theta}\right)$$

Then the area under one arc of the cycloid (i.e. where $$0 \le \theta \le 2 \pi$$) is $$3 \pi a^2$$.

That is, the area under one arc of the cycloid is three times the area of the generating circle.

Proof
Let $$A$$ be the area under of one arc of the cycloid. Then:

The area required is defined by:

$$ $$

But:
 * $$\frac{\mathrm{d}{x}}{\mathrm{d}{\theta}} = a \left({1 - \cos \theta}\right)$$

and so:

$$ $$ $$ $$ $$