Gaussian Integral, the Jacobian and the Gamma Function

To evaluate the Gaussian Integral requires the use of double integrals and the Jacobian for the change of variables from rectangular to polar coordinates. The following YouTube video gives a clear account of how this integral is evaluated:

The following video deals with the gamma function for value 1/2 (=1/2!) and the use of the above integral technique to evaluate it.

The final video is by the author of the second video and is an introduction to the gamma function. I'm just skimming the surface of things here but it's a start.