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Explain $\iint \mathrm dx\mathrm dy = \iint r \mathrm d\alpha\mathrm dr$

I'm reading the proof of Gaussian integration. When we change to polar coordinates, why do we get an "extra" r in there?

\begin{align} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-(x^2+y^2)}\ dx dy &= \int_0^{2\pi} \int_0^{\infty} e^{-r^2}r\ dr\ d\theta\\ \end{align}

I've looked at a few different proofs:

but none explain this step fully enough for me to really see why this happened.

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    This question has already been addressed and can be found [here](http://math.stackexchange.com/questions/37044/explain-iint-mathrm-dx-mathrm-dy-iint-r-mathrm-d-alpha-mathrm-dr).2011-07-06
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    The first link in the question should be changed to http://www.math.uconn.edu/~kconrad/blurbs/analysis/gaussianintegral.pdf2012-04-04

2 Answers 2