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I'm trying to evaluate the standard Gaussian integral

[tex]\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}[/tex]

The standard method seems to be by i)squaring the integral, ii)then by setting the product of the two integrals equal to the iterated integral constructed by composing the two integrals, iii)using Fubini's theorem to turn this into an area integral and iv)then using Fubini's theorem again to turn this back into an iterated integral, this time in polar coordinates.

Of these, (ii) seems impossible to me. Why would the iterated integral be equal to the product of the two integrals taken separately? Even if this were the case, this does not seem like a result so trivial that you could use it without any justification. Are there other ways to evaluate the integral? Thanks.

Molu

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# Evaluating the Gaussian integral

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