I have a math assignment. The question statement is "Application of change of variables in double and triple integral. Give the relation between e and pi using this." I did a web search and went through my textbook (Thomas' Calculus) but I'm unable to find a relation between $e$ and $\pi$ in the context of multiple integrals.
Also, what are the applications of change of variables in multiple integrals?
One thing which comes to my mind is evaluating the integral $$ I= \int _0^{\infty}e^{-x^2}dx =\sqrt {\pi}/2$$
Note that
$$ I^2 = \int _0^{\infty} \int _0^{\infty}e^{-x^2}e^{-y^2}dydx =$$
$$ \int _0^{\pi /2} \int _0^{\infty}e^{-r^2}rdr d\theta =\pi/4$$
The trick is to change the dummy variable from $x$ to $y$ and multiply to get a double integral which is changed to polar coordinates and easily evaluated.
The result of $\sqrt {\pi}/2 $ is found by taking the square root of the double integral .