I want to use Green's theorem for computing the area of the region bounded by the $x$-axis and the arch of the cycloid:
$ x = t- \sin (t),\;\;\; y = 1 - \cos (t),\;\; 0 \leq t \leq 2\pi $
So basically, I know the radius of this cycloid is 1. And to use Green's theorem, I will need to find $Q$ and $P$.
$\int_C P\,dx + Q\,dy = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA$