The  Gaussian integral or otherwise known as the Euler-poisson integral of the Gaussian function e^(-x^2) over  the entire real line.
The integral is:


The Gaussian integral can be solved analytically through the methods of multivariable calculus.  
i.e there is no elementary indefinite Integral for but the definite
integral
Can be evaluated. 
Thus the Gaussian integral can be integrated or computed by two methods Namely :
1)Polar Coordinates 
2)Cartesian Coordinates 
We will begin with the Polar Coordinates

1)Polar Coordinates 
The Gaussian integral can be computed using the idea of poisson by using the property that 



Now let On the plane R². Now compute it's integral in two ways ;
i) by double integral in the Cartesian Coordinates system, it's integral is a square. 


ii) by shell integration (this is simply a case of double integral in Polar Coordinates). It's integral is computed to be Ï€. 
By comparing these two computations, we get

The factor of r here comes from the transform to Polar Coordinates (rdrdθ)  is the standard measure on the plane, expressed in Polar Coordinates. And the substitution involves taking S= -r², so ds= -2rdr combining these yields 

2) by Cartesian Coordinates 
Using Laplace technique which dates back to 1812, we simply assume
 y=xs
dy/ds=x =>dy=xds
As lim y=±∞
This enables us to use the fact that e^(-x^2) is an even function and therefore the integral over all real numbers is just twice the integral from zero(0) to infinity(∞) That is 
and that brings the question to an End.