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 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 $A$ is defined by:

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

and so:

Also see

 * Length of Arc of Cycloid

Historical Note
This was discovered in $1644$ by.

had previously estimated the area as being approximately $3$ times the area of the generating circle.

It is believed that found this result earlier than, but lost priority when he failed to publish his proof.