how do you prove that the value of int(e^(-x^2)) from -infinity to +infinity is root pi?

Question

how do you prove that the value of int(e^(-x^2)) from -infinity to +infinity is root pi?

what are the steps involved? any name to this equation?

Answer 1

Consider the double integral
exp(-x^2)*exp(-y^2) dx dy

Convert to polar cords:
exp(-r^2) * r*dr*dθ

Then integrate from 0 < r < inf
and 0 < θ < 2π

Let u = r^2, du = 2r*dr
Now integrate
1/2 exp(-u) du dθ
where 0 < u < inf

You get 1/2 * (1-0) * (2π)
= π

This is the square of the original integral.