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Fix $a\in [0,1]$ and consider the closed curve

$ C_a:= \{ (x,y) : \sin(x) \sin(y) = a,\ (x,y)\in [0,\pi]\times[0,\pi]\}. $

See contour plot of function $\sin(x) \sin(y)$. I am interested in computing the area of the enclosed by $C_a$ as a function of $a$. Is there a simple analytical form?

enter image description here

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A partial answer:

First, change your variables $x \to \pi/2 - x$ and $y \to \pi/2 - y$ so your integral becomes:

$$4 \int\limits_{x = 0}^{\cos^{-1} a} \cos^{-1}\left( \frac{a}{\cos x} \right)\ dx$$

which should be clear from the plot:

enter image description here

Unfortunately, there seems to be no closed form for arbitrary $a$. (Of course you can perform the integration numerically for an assigned value of $a$.)

You seem to be Mathematica literate, so use this instead of your code and insert your value for $a$ in the Contour level:

ContourPlot[Cos[x] Cos[y],
 {x, -π/2, π/2},
 {y, -π/2, π/2},
 AxesOrigin -> {0, 0},
 Axes -> True,
 Contours -> {0.1}]
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