Hi

I am trying to learn the proof for the value of the Gaussian Integral posted here:

Gaussian integral - Wikipedia, the free encyclopedia
under 'By Cartesian Coordinates"

I understand the basics of double integration, but I don't quite understand this proof.

I see that the function is an even function...but what is the significance of saying "Let x=ys and dy=xds"

I also cannot see how

$\displaystyle 4\int^\infty_0\int^\infty_0e^{-(x^2+y^2)}dydx$

arises when

$\displaystyle 2\int^\infty_0e^{-x^2}dx$ is squared.

And...I also do not understand how the 's' variable arises later on into the proof.

I would greatly appreciate if these questions were answered and it would be extremely appreciated if a quick run through of this proof was posted.

Thank you very much.