Multiple integrals stack one integral sign per variable
When a function depends on several real inputs—say f(x,y) or f(x,y,z)—its definite integral over a region is called a multiple integral, the higher-dimensional cousin of area-under-a-curve. Integrals over solid regions are labelled triple integrals; n variables bring n nested signs over a domain D.
A multiple integral is the definite integral of a function of several real variables over a region; over the plane it is called a double integral and over three-dimensional space a triple integral. Just as a one-variable integral of a positive function gives the area between its graph and the x-axis, a double integral of a positive function gives the volume between the surface z = f(x, y) and the plane beneath it. By convention a double integral carries two integral signs and a triple integral three.
Because antiderivatives are defined only for functions of one variable, the indefinite integral does not carry over directly. Instead the Riemann definition starts with a half-open box in n dimensions, cuts each side into subintervals closed on the left and open on the right, and forms a sum of function values times the measure of each small sub-box. When that sum approaches a single value as the largest sub-box diameter shrinks to zero, the function is Riemann integrable and the value is its integral.
To integrate over an arbitrary bounded region, the function is extended to a surrounding box by setting it to zero outside the original domain. Multiple integrals share linearity, monotonicity and other properties with ordinary integrals, and under suitable conditions Fubini's theorem makes the result independent of the order of integration, which for continuous functions justifies the usual tactic of reducing the problem to a chain of one-variable integrals.
A constant integrand c yields c times the measure of the domain, so integrating 1 over a planar region gives its area and over a solid gives its volume. For example, integrating the constant 2 over the rectangle where x runs from 2 to 4 and y from 3 to 6 gives 2 times an area of 6, or 12.
Source: Multiple integral
