Do you know the Gaussian Integral?

See this method

The expression in form,

plays an important role in mathematics and statistics. It is said that the integral of this function was first observed, associated with the meaning of the error function, by the mathematician Abraham de Moivre and was later solved precisely, in its improper form, by Gauss. Such a solution is what we will present here.

Then be the following definition:

then we will determine the value of F.

Let us initially observe that the function f has the property of being an even function, that is, in its representation in the Cartesian plane it will present symmetry. The consequence of this is that we can rewrite:

Likewise, this observed symmetry also propagates in a three-dimensional representation of an analogous function:

or

as suggested in the figure and in any radial direction.

Therefore, this also suggests that we can now deduce that our task is to first determine the volume beneath the three-dimensional surface of the figure, and particularly to use a polar coordinate system in the xOy plane. In other words, we will determine

Since we will use polar coordinates, then variable transformation rules must be applied, that is, if

in polar coordinates it will be

where J is the Jacobian of the transformation, which in the present case, we have J = r, and therefore, we can now rewrite

or

expression whose result is trivial and given by

or better:

this can also be written as,

It is also trivial to realize that, asymptotically

and therefore,

then