area under Gaussian curve

Theorem.

The area between the curve   $y=e^{-x^2}$  and the $x$-axis equals $\sqrt{\pi }$,  i.e.

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-x^2}𝑑x=\sqrt{\pi }.$$

Proof.  The square of the area is

${\left({\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{e}^{-x^2}𝑑x\right)}^2$ $=\underset{a\to \mathrm{\infty }}{lim}{\left({\displaystyle {\int }_{-a}^a}{e}^{-x^2}𝑑x\right)}^2$

$=\underset{a\to \mathrm{\infty }}{lim}{\displaystyle {\int }_{-a}^a}{e}^{-x^2}𝑑x\cdot {\displaystyle {\int }_{-a}^a}{e}^{-y^2}𝑑y$

$=\underset{a\to \mathrm{\infty }}{lim}{\displaystyle {\int }_{-a}^a}{\displaystyle {\int }_{-a}^a}{e}^{-(x^2+y^2)}𝑑x𝑑y$

$=\underset{R\to \mathrm{\infty }}{lim}{\displaystyle {\int }_0^R}{\displaystyle {\int }_0^{2\pi }}{e}^{-r^2}r𝑑r𝑑\phi$

$=\underset{R\to \mathrm{\infty }}{lim}2\pi {\displaystyle {\int }_0^R}{e}^{-r^2}r𝑑r$

$=-\pi \underset{R\to \mathrm{\infty }}{lim}\underset{0}{\overset{R}{/}}{e}^{-r^2}$

$=\pi \underset{R\to \mathrm{\infty }}{lim}(1-e^{-R^2})=\pi .$

Here, the limit of the double integral over a square has been replaced by the limit of the double integral over a disc, because both limits are equal.  That both limits are equal can be demonstrated by the elementary

$$0\le {\int }_{-a}^a{\int }_{-a}^a{e}^{-(x^2+y^2)}𝑑x𝑑y-{\int }_0^a{\int }_0^{2\pi }{e}^{-r^2}r𝑑r𝑑\phi \le \underset{greatestvalue}{\underbrace{e^{-a^2}}}\cdot \underset{area}{\underbrace{(4a^2-\pi a^2)}}=(4-\pi )\cdot \frac{a^2}{e^{a^2}},$$

and  $\frac{a^2}{e^{a^2}}\to 0$  when  $a\to \mathrm{\infty }$  (see growth of exponential function).

Remark.  Since $e^{-x^2}$ is an even function,

${\displaystyle {\int }_0^{\mathrm{\infty }}}{e}^{-x^2}dx={\displaystyle \frac{\sqrt{\pi }}{2}}\cdot$

Title	area under Gaussian curve
Canonical name	AreaUnderGaussianCurve
Date of creation	2013-03-22 15:16:36
Last modified on	2013-03-22 15:16:36
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	22
Author	pahio (2872)
Entry type	Theorem
Classification	msc 26B15
Classification	msc 26A36
Synonym	Gaussian integral
Synonym	area under the bell curve
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