Multiple Integral


Also found in: Wikipedia.

multiple integral

[′məl·tə·pəl ′int·ə·grəl]
(mathematics)
An integral over a subset of n-dimensional space.

Multiple Integral

 

an integral of a function defined on some region in a plane and in three-dimensional or n -dimensional space. The corresponding multiple integrals are referred to as double integrals, triple integrals, and n-tuple integrals, respectively.

Let the function f(x, y ) be defined on some region D of the plane xOy. Let us divide D into n subregions di whose areas are equal to si, choose a point (ξi, ηi) in each subregion di (see Figure 1), and form the integral sum

If as the maximal diameter of the subregions d, decreases without bound the sums S have a limit independent of the choice of the points (ξi, ηi), then this limit is called the double integral of the function f(x, y) over the region D and is denoted by

∫ ∫Df (x,y) ds

A triple integral and, in general, an n -tuple integral are defined analogously.

Figure 1

In order for the double integral to exist, it is sufficient that, for example, the region D be a closed (Jordan) measurable region and that the function f(x, y) be continuous throughout D. Multiple integrals possess a number of properties similar to those of ordinary integrals. In order to calculate a multiple integral we reduce it to an iterated integral. Green’s formulas and the Green-Ostrogradskii theorem can be used in special cases to reduce multiple integrals to integrals of lower dimension. Multiple integrals find wide application. Volumes of bodies, as well as masses, static moments, and moments of inertia ( of bodies, for example) are expressed using multiple integrals.

References in periodicals archive ?
As part of the 2012 startup project the turbo blower control system required upgrading, and Cordell provided the project management, design, manufacture testing and, installation & commissioning resources on multiple integral elements.
Tseng, On certain multiple integral inequalities related to Hermite-Hadamard inequalities, Utilitas Math.
Buse, Refinements of Hadamard's inequality for multiple integrals, Utilitas Math.
on multiple integral marketing promotions on film and game releases.
in the form of multiple integral and proved the following integral inequality
The invitation is restricted to certain eligible institutional investors and bonds may only be tendered for purchase in a minimum principal amount of[euro]1,000,000 and multiple integrals of[euro]1 in excess thereof.
The second volume follows the elementary material of the first with discussion on functions of several variables and their derivatives, multiple integrals, differential equations, functions of a complex variable, and other topics.
G [2] evaluated multiple integrals of a bounded function of n-real variables where the rule has been determined for n = 2,3 numerically and an asymptotic error for n = 2 is verified.
Using the above theorem, one can convert variational problems with multiple integrals into problems with curvilinear integrals and reciprocally.
Some third of the material is concerned with biographical and other contextual issues, while the bulk of the selections focus on particular aspects of Euler's contributions to mathematics, including infinite series, the zeta functions, Euler's constant, differentials, multiple integrals, the calculus of variations, the pentagonal number theorem, quadratic reciprocity, and the fundamental theorem of algebra.
The issue will be sold as current interest revenue bonds in denominations of $500, or multiple integrals, and capital appreciation revenue bonds in denominations of $200, or multiple integrals.
Topics include special matrices, non-negative matrices, special products and operators, Jacobians, partitioned and patterned matrices, matrix approximation, matrix optimization, multiple integrals and multivariate distributions, linear and quadratic forms, etc.

Full browser ?