Improper multiple integral

Remarks

For non-negative functions, the limit does not depend on the choice of the exhausting sequence of bounded sets. For functions of both signs one usually requires that $\int |f|$ is finite. Example: in polar coordinates, $\iint_{\mathbb{R}^2} e^{-(x^2 + y^2)} \, dA = \lim_{a \to \infty} \int_0^{2\pi} \int_0^a e^{-r^2} r \, dr \, d\theta = \pi$. Example with a singularity: $\iint_{\|x\| \le 1} \|x\|^{-s} \, dA$ is finite for $s < 2$ and infinite for $s \ge 2$, by the same polar computation. An improper double integral can fail to exist even when both iterated integrals exist, if $\int |f| = \infty$.