Nonlinear Systems & Linearization · Superchallenge problem

For the undamped pendulum $x' = y,\; y' = -\sin x$, the closed orbits around the equilibrium $(0, 0)$ fill an "eye"-shaped region bounded by the two separatrices joining the saddles $(-\pi, 0)$ and $(\pi, 0)$. Find the area of this region.