Let $D$ be the parallelogram on the $(x,y)$ plane ($z=0$) that comes from the intersection of the lines: $y=x$, $x = \pi$, $y = x + \pi$ and $x=0$.
Compute the following integral:
$$ \int_S \sqrt{ 1 + 2 \cos^2(y-x) } dS $$
where $S$ is the surface described by the equation $z = \sin(y-x)$ projected on $D$.