What other definite integrals can be computed in a manner similar to $\int_{-\infty}^\infty e^{-x^2}dx$? The technique for computing $\int_{-\infty}^\infty e^{-x^2} dx=\sqrt{\pi}$ by computing the integral squared using polar coordinates is well known. Are there any other integrals that can be computed in a similar way? My question is intentionally vague--any techniques however tangentially related are of interest!
(EDIT 1: To be more specific, what I am imagining is whether it is possible that some definite integrals appear algebraically out of a computation of a double or triple integral in some coordinate system. Or, if there are some heuristic arguments that this probably impossible. Or if someone has a better imagination as to a computation being similar to the example I gave is also good.)
(EDIT 2: It would also be interesting if someone could example how one might come up with the computation in the original example instead of it being seen as an after the factor sort of thing.)
 A: That is essentially the only integral that this trick is good for.  See
http://www.unf.edu/~dbell/Poisson.pdf
A: Apparently others have already done it more than 150 years ago ! The following Wikipedia articles about certain two-dimensional and three-dimensional geometric shapes characterized by algebraic equations similar to $X^n+Y^n(+Z^n)=R^n\iff x^n+y^n(+z^n)=1$ contain the mathematical expressions of such generalized polar coordinates: astroids ($2$D), super ellipses ($2$D), super formulas ($2$D), super ellipsoids ($3$D), super quadrics ($3$D), super toroids ($3$D), etc. They are rational powers of trigonometric functions, whose areas and volumes are expressible in terms of the Beta and Gamma functions. As usual, my ideas are a century too late, and a buck too short. :-) I hope that you'll find these links useful, and sorry for posting this as an answer, but it was too long to fit into the size of a single comment.
A: Complex Analysis: Evaluate $\int_{-\infty}^{\infty}\frac{x^2}{1+x^4}\ dx$
Let $f(z)=\frac{z^2}{1+z^4}$.
$(1) \lim_{\ r\to\infty}\int_{-r}^{r}\ \frac{z^2}{1+z^4}\ dz$
$(2) \lim_{\ r\to\infty}\int_{-r}^{r}\ \frac{z^2}{1+z^4} dz = \int_{c}f(z)\ dz -\int_{c_r} f(z)\ dz$. (notice $z$ is on the real axis)
Here we take $c$ to be the semi circle of radius $r$ and $c_r$ to be the arc of $c$.
$(3)$ By the Estimation Lemma (or M-L lemma) $\int_{c_r} f(z)\ dz = 0$.
$(4)$ Now we can apply a special case of the Residue Theorem to evaluate,  $\int_{c}f(z)\ dz$.
$(5)$ So by R.T,  $\int_{c}f(z)\ dz = \sum Res_{z=z_0} f(z)$.
We can finish up from here, but I don't know about your complex variables background, but yes this is another interesting technique that complex analysis presents you with.
