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Kind of intentionally obtuse since they used eₑ as a variable and eₑₑ as another variable, and used (e-e) as an exponent a few times, which is basically the equivalent of multiplying by 1 in a fancy way. The first and last term also perfectly cancel out.
The same integral written in a saner form is:
integral from -e^e to e^e of (integral from -e^e to e^e of e^-(x^2+y^2)dy)dx
Wait… that’s not an approximation at all! That equals exactly pi. If I understand the math correctly, it’s effectively a formula for the area of a unit circle.
That should be an approximation. To get exactly pi the range of both integrals should be from minus infinity to infinity like this. It’s the integral of the 2D Gaussian, which is fairly known.
