PlanetMath: area under Gaussian curve

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area under Gaussian curve

(Theorem)

Theorem 1 The area between the curve $y = e^{-x^2}$ and the

-axis equals $\sqrt{\pi}$ , i.e.

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

$\includegraphics{gaussian-curve.eps}$

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Here, the limit of the double integral over a square has been replaced by the limit of the double integral over a disc, because both limits are equal.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

$\displaystyle 0 \leq \int_{-a}^a \int_{-a}^a\!e^{-(x^2 + y^2)} dx dy - \int_0... ...nderbrace{(4a^2\!-\!\pi a^2)}_{area} = (4\!-\!\pi)\!\cdot\!\frac{a^2}{e^{a^2}},$

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,

$\displaystyle \int_0^\infty e^{-x^2} dx=\frac{\sqrt{\pi}}{2}\:\cdot$

"area under Gaussian curve" is owned by pahio. [ full author list (4) ]

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See Also: substitution notation, proof that normal distribution is a distribution, probability distribution function, error function, evaluating the gamma function at 1/2, normal random variable, table of probabilities of standard normal distribution

Other names:

Gaussian integral, area under the bell curve

Keywords:

double integral, polar coordinates

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Cross-references: even function, growth of exponential function, disc, integral, limit, square, proof, curve, area
There are 5 references to this entry.

This is version 19 of area under Gaussian curve, born on 2005-05-17, modified 2008-08-15.
Object id is 7065, canonical name is AreaUnderGaussianCurve.
Accessed 13408 times total.

Classification:

AMS MSC:	26A36 (Real functions :: Functions of one variable :: Antidifferentiation)
	26B15 (Real functions :: Functions of several variables :: Integration: length, area, volume)

Pending Errata and Addenda

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