Multiple Integration

The multiple integral is a generalization of the definite integral to functions of more than one real variable, for example, f(x, y) or f(x, y, z). Integrals of a function of two variables over a region in R2 are called double integrals, and integrals of a function of three variables over a region of R3 are called triple integrals


Introduction

Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the x-axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function (on the three-dimensional Cartesian plane where z = f(x, y)) and the plane which contains its domain. (The same volume can be obtained via the triple integral—the integral of a function in three variables—of the constant function f(x, y, z) = 1 over the above-mentioned region between the surface and the plane.)[1] If there are more variables, a multiple integral will yield hypervolumes of multidimensional functions.

Multiple integration of a function in n variables: f(x1, x2, ..., xn) over a domain D is most commonly represented by nested integral signs in the reverse order of execution (the leftmost integral sign is computed last), followed by the function and integrand arguments in proper order (the integral with respect to the rightmost argument is computed last). The domain of integration is either represented symbolically for every argument over each integral sign, or is abbreviated by a variable at the rightmost integral sign.