Integrating a Gaussian

The work on wave packets has led me to the Gaussian function exp(-ax^2). Here I plotted for a = 0.01. I was interested that a had to be so small to spread the curve out. Integrating this means finding the area under the curve. It tells you how to do the integration here http://www.umich.edu/~chem461/Gaussian%20Integrals.pdf The big two tricks seem to be to first square the integral and then use this to switch to polar coordinates. I was struggling with the area element dA - the small increase in area. In Cartesian coordinates that is dA = dxdy but in polar coordinates it becomes dA = rdrd(theta). This website has a picture and explains all http://citadel.sjfc.edu/faculty/kgreen/vector/block3/jacob/node4.html