improper multiple integral
An ordinary multiple integral assumes a bounded region and a bounded integrand. An improper multiple integral is what you do when one of those fails: the region runs off to infinity (integrating over the whole plane or all of space), or the integrand blows up somewhere inside the region (a density that becomes infinite at a point, like a 1/distance potential at its source). As with improper single integrals, you tame the difficulty by taking a limit — you integrate over a growing or shrinking safe region and watch where the value heads.
The standard method is exhaustion. To integrate over an unbounded region, integrate over a bounded piece — a disk of radius R, or a box, or a region that stops short of the singular point — and then let that piece swell to fill the whole region (R going to infinity) or close in on the bad point. If the limit exists and is finite, the improper integral converges to it. A crucial and reassuring fact special to multiple integrals: when the integrand has one sign (or is absolutely integrable), the limit does not depend on the shape of the exhausting regions — disks, squares, and any other reasonable expanding family all give the same answer. This is exactly what makes the polar-coordinate evaluation of the multidimensional Gaussian integral legitimate: you may compute it over expanding disks and trust the result.
Improper multiple integrals are everywhere in physics and probability, where fields, potentials, and probability densities are naturally defined over all of space and often have a singularity at a source. The honest warning is the flip side of the reassuring fact: for an integrand that changes sign and is not absolutely integrable, the answer can depend on how you exhaust the region — squares and disks can give different limits, just as a conditionally convergent series can be rearranged to sum to anything. So convergence of an improper multiple integral really means absolute convergence (the integral of the absolute value is finite); when that fails, no single value deserves to be called the integral.
The integral of e^{-(x^2+y^2)} over the whole plane is improper (unbounded region). Exhaust by disks of radius R: integral over theta from 0 to 2pi, r from 0 to R of e^{-r^2} r dr d-theta = pi(1 - e^{-R^2}), which tends to pi as R goes to infinity.
Because the integrand is positive, the disk-exhaustion limit is the honest value pi — the foundation of the Gaussian integral.
For a sign-changing, non-absolutely-integrable integrand the limit can depend on the exhausting shapes (disks vs squares), so the integral has no well-defined value. Genuine convergence of an improper multiple integral means absolute convergence.