Multiple integrals

A single integral slices an interval and adds up contributions along a line. A double integral does the same over a region of the plane: cut it into small patches, multiply each patch's area by the value of the function there, and add. The evaluation trick is that this reduces to two ordinary integrals done in succession — integrate with respect to one variable while treating the other as constant, then integrate the result.

The limits carry the geometry, and this is where the real work is. Over a rectangle all four limits are constants. Over a triangle or a disc the inner limits must depend on the outer variable, because the region's extent in one direction changes as you move along the other. Writing those limits correctly is usually harder than the integration itself, and sketching the region first is not optional.

Two cautions. Swapping the order of integration is often the difference between an easy problem and an impossible one, but the limits must be rewritten for the new order, not simply exchanged where they stand — Fubini's theorem promises the same answer, not that the same symbols still apply. And a double integral is not "the thing that gives volume". Integrate the constant 1 and you get area; integrate a height and you get volume; integrate a density and you get mass; integrate a probability density and you get probability. The integrand decides what is being accumulated, which is exactly why multiple integrals appear wherever a quantity is spread over a region rather than concentrated at a point.