In my calculus textbook, a squeeze technique is used to prove the result via double integrals in polar coordinate.
In probability theory the important Gaussian Distribution often leads to the integral .
In case you ever wondered where it comes from, here (informal but nice*) is a non-complex analysis demonstration.
We begin by considering the integral,
This is the integral over the entire plane .
Now there are two ways to get the entire plane.
Method 1: For we can create a rectangle . Now we make . That infinite rectangle will take the entire plane.
In other words, .
Now, the important realization is that this double integral has "seperated variables" meaning and we can write . Hence, (because it cannot be negative).
Method 2: We notice the expression in the double integral and try a polar coordinates substitution. If we let , we can let and . In other words, we draw a circle at the orgin with infinite radius, and this will take on values on the entire plane.
Thus, we end up with after a change to polar coordinates,
Thus, which implies .
*)I am sure it can be made formal. But I do not know how do it because I am not familar with the theory of real multiple integration.