multiple integral

A single integral adds up thin strips under a curve to get an area. A multiple integral does the same trick in higher dimensions. Picture a lumpy tent pitched over a rectangular field: to find the volume of air trapped underneath, you chop the field into tiny tile-sized squares, multiply each tile's area by the tent's height above it, and add all those little boxes together. That total is a double integral — accumulation spread over a region instead of along a line.

For a function f(x, y) over a region R, the double integral is written double integral over R of f(x, y) dA, and it is defined the same rigorous way as a single integral: a limit of Riemann sums as the tiles shrink toward zero. In practice you usually evaluate it as an iterated integral, integrating one variable at a time: integral from c to d ( integral from a to b f(x, y) dx ) dy. A triple integral integral integral integral f dV extends this to solid three-dimensional regions.

Multiple integrals are how we accumulate any quantity spread through space: volume under a surface, the mass of an object from its density, the total charge in a region, an average over an area, or a center of mass. The order of integration can be swapped (for well-behaved functions, by Fubini's theorem), and choosing a clever order — or switching to polar or spherical coordinates — often turns an impossible-looking integral into a routine one.