Proof for Area under Gaussian distribution

This section concerns itself with finding the integral of a Gaussian Distribution

\int_{- \infty}^{\infty} e^{- x^{2}} d x = C

Find $C$ .

Proof

Lets consider adding another dimension.

f (x, y) = e^{- x^{2} - y^{2}} = e^{- x^{2}} \cdot e^{- y^{2}} = e^{- r^{2}}

where $r^{2} = x^{2} + y^{2}$ . Let $r$ be the radius of the circle. Thus all points a distance $r$ from the origin on $e^{- x^{2} - y^{2}}$ would have the same value. We can graph out the function against $x$ and $y$

We can find the volume under $f (x, y)$ by summing the area under the red line (points of equidistant from the origin) for all values $r$ . Unroll the area under the red line to form a rectangle, with length $2 π r$ and height $e^{- r^{2}}$ , integrate with respect to $r$ .

\begin{aligned} V & = \int_{0}^{\infty} 2 π r e^{- r^{2}} d r \\ = π \int_{0}^{\infty} 2 r e^{- r^{2}} d r \\ = π {[- e^{- r^{2}}]}_{0}^{\infty} \\ = π \end{aligned}

We can express the volume as a double integral as below, where the inner integral integrates in the $x$ axis, keeping $y$ as a constant. I like to think of the integral as infinitely slicing $f (x, y)$ along the $x$ axis and finding the area bounded by $f (x, y)$ in each slice, expressing it as some value. Next the outer integral sums up all these areas by integrating in the $y$ direction, thus obtaining the volume $(V)$ under $f (x, y)$ .

\begin{aligned} V & = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} e^{- x^{2} - y^{2}} d x d y \\ = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} e^{- x^{2}} e^{- y^{2}} d x d y \\ = \int_{- \infty}^{\infty} e^{- y^{2}} \int_{- \infty}^{\infty} e^{- x^{2}} d x d y [We can factor e^{- y^{2}} as it is a constant] \\ = \int_{- \infty}^{\infty} e^{- y^{2}} C d y \\ = C \int_{- \infty}^{\infty} e^{- y^{2}} d y \\ = C^{2} \end{aligned}

Thus

\begin{aligned} V = C^{2} & = π \\ \int_{- \infty}^{\infty} e^{- x^{2}} d x & = C \\ = \sqrt{π} \end{aligned}

Thus

$\int_{- \infty}^{\infty} e^{- x^{2}} d x = \sqrt{π}$

Source: https://www.youtube.com/watch?v=cy8r7WSuT1I