I'm trying to find the area of an irregular domain that is bounded by $x = c$, $y = c$, and $c = -A\sin(x/2)\sin(y/2)+\cos(x/2)\cos(y/2)$, where A can vary in the range [-1,1], and x and y are only defined over $x\in[0,\pi]$ and $y\in[0,\pi]$. I need to know the area as a function of the parameter A. I was thinking of trying to use green's theorem and just integrating around the border of the region, but in order to do so I need a smart parameterization of $c = -A\sin(x)\sin(y)+\cos(x)\cos(y)$. Can anyone think of a good way to parameterize this function or the boundary in general? Or any better ideas of how to get the area of this region?
In practice I express c as $c = \cos(z/2)$ so that the solution to the equation $\cos(z/2) = -A\sin(x/2)\sin(y/2)+\cos(x/2)\cos(y/2)$ when c is a constant are level sets. The equations $x = c$, $y = c$, and $c = -A\sin(x/2)\sin(y/2)+\cos(x/2)\cos(y/2)$, as well as the domain boundaries, $x = 0$, $x = \pi$, $y = 0$, and $y = \pi$, subdivide the domain into a number of regions and I'm actually interested in the area of each of the regions as a function of the parameters A and c, an example of a representative situation is given here, where you can see 7 different regions whose area I am interested in.
Thank you so much for your help.
if the "here" link doesn't work: