double integral

A single integral measures the area under a curve over an interval; a double integral does the same job one dimension up. Picture a region R lying flat in the xy-plane and a surface z = f(x, y) floating above it. The double integral of f over R is the signed volume trapped between that surface and the plane — and if f is just the constant 1, the answer is simply the area of R itself. It answers the question: when a quantity is spread over a two-dimensional region, how much of it is there in total?

It is built exactly the way the single Riemann integral is built, only with little tiles instead of little strips. Chop R into many small rectangles of area delta-A = delta-x times delta-y, pick a sample point in each one, multiply the height f at that point by the tile's area, and add up all those thin boxes f(x, y) delta-A. As the tiles shrink to nothing, the sum tends to a limit, written as the double integral over R of f(x, y) dA. The symbol dA is the area element; in plain Cartesian coordinates it is dx dy. The integral exists whenever f is, for instance, continuous on a reasonable region.

Double integrals are how you total any density over a flat region: mass of a metal plate with surface density rho(x, y), total electric charge over a region, the probability that a pair of random variables lands inside a set (with f the joint density), or the average value of a temperature field. In practice you almost never compute the limit of sums directly — you turn the double integral into two ordinary integrals done one after the other (an iterated integral), which Fubini's theorem licenses. The whole craft of double integration is choosing the right region description and the right order so those two single integrals are doable.