area under Gaussian curve


The area between the curve   y=e-x2  and the x-axis equals π,  i.e.


Proof.  The square of the area is

(-e-x2𝑑x)2 =lima(-aae-x2𝑑x)2

Here, the limit of the double integral over a square has been replaced by the limit of the double integral over a disc, because both limits are equal.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


and  a2ea20  when  a  (see growth of exponential function).

Remark.  Since e-x2 is an even functionMathworldPlanetmath,

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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