multiple integral

A single integral adds up a function over an interval — think of the area under a curve. A multiple integral does the same over a region in two, three, or more dimensions: it accumulates the values of a function across a flat plate, a solid body, or a higher-dimensional region. Picture slicing the region into tiny tiles, weighing each by the function, and summing.

Formally, the Riemann multiple integral of f over a region D in R^n is the limit of Riemann sums: partition D into small cells, multiply the value of f in each cell by the cell's n-dimensional volume, sum, and refine the partition. When this limit exists independently of the choices, f is integrable over D and the limit is written as the integral of f over D, with volume element dx (or dV).

For a non-negative f over a planar region, the double integral is the volume under the graph; over a solid, the triple integral can give mass when f is a density. Bounded functions that are continuous except on a set of measure zero are integrable — and in practice we almost never compute the limit directly. Instead we turn the multiple integral into iterated single integrals via Fubini's theorem.