area under Gaussian curve

Theorem.

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

 $\int_{-\infty}^{\infty}e^{-x^{2}}\,dx=\sqrt{\pi}.$

Proof.  The square of the area is

 $\displaystyle\bigg{(}\int_{-\infty}^{\infty}e^{-x^{2}}\,dx\bigg{)}^{2}$ $\displaystyle=\lim_{a\to\infty}\bigg{(}\int_{-a}^{a}e^{-x^{2}}\,dx\bigg{)}^{2}$ $\displaystyle=\lim_{a\to\infty}\int_{-a}^{a}e^{-x^{2}}\,dx\,\cdot\int_{-a}^{a}% e^{-y^{2}}\,dy$ $\displaystyle=\lim_{a\to\infty}\int_{-a}^{a}\int_{-a}^{a}e^{-(x^{2}+y^{2})}\,% dx\,dy$ $\displaystyle=\lim_{R\to\infty}\int_{0}^{R}\!\int_{0}^{2\pi}e^{-r^{2}}r\,dr\,d\varphi$ $\displaystyle=\lim_{R\to\infty}2\pi\!\int_{0}^{R}e^{-r^{2}}r\,dr$ $\displaystyle=-\pi\!\lim_{R\to\infty}\!\operatornamewithlimits{\Big{/}}_{\!\!% \!0}^{\,\quad\,\,R}e^{-r^{2}}$ $\displaystyle=\pi\!\lim_{R\to\infty}(1-e^{-R^{2}})\;=\;\pi.$

Here, the limit of the double integral over a square has been replaced by the limit of the double integral over a disc, because both limits are equal.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

 $0\leq\int_{-a}^{a}\int_{-a}^{a}\!e^{-(x^{2}+y^{2})}\,dx\,dy-\int_{0}^{a}\!\int% _{0}^{2\pi}\!e^{-r^{2}}r\,dr\,d\varphi\leq\underbrace{e^{-a^{2}}}_{greatest\,% value}\!\cdot\,\underbrace{(4a^{2}\!-\!\pi a^{2})}_{area}=(4\!-\!\pi)\!\cdot\!% \frac{a^{2}}{e^{a^{2}}},$

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

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

 $\displaystyle\int_{0}^{\infty}e^{-x^{2}}\,dx=\frac{\sqrt{\pi}}{2}\>\cdot$
 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 Related topic SubstitutionNotation Related topic ProofThatNormalDistributionIsADistribution Related topic Distribution Related topic ErrorFunction Related topic EvaluatingTheGammaFunctionAt12 Related topic NormalRandomVariable Related topic TableOfProbabilitiesOfStandardNormalDistribution Related topic ApplyingGeneratingFunction Related topic FresnelFormulas