# Multiple Integral

(redirected from Double integrals)

## multiple integral

[′məl·tə·pəl ′int·ə·grəl]
(mathematics)
An integral over a subset of n-dimensional space.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## 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.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
Here the data dispersion of simple and double integrals during the unweight and propulsion phases of the countermovement jump described a region of great dispersion in 1-7 Hz, and described a region of divergence (variance increment) after 14 Hz.
Chen, "Ostrowski type inequalities on time scales for double integrals," Acta Applicandae Mathematicae, vol.
One of the questions included in the open-ended questionnaire was: "What are the differences between your views before and after the implementation of the flipped classroom approach in the study of double integrals? Please explain."
An Ostrowski type inequality for double integrals and applications for cubature formulae.
On the base of the abovementioned, it is possible to express the double integrals in (B.7) and (B.8) as follows:
Even if this method can be immediately extended to the analysis of the radiation and the scattering by a perfectly conducting rectangular plate in a homogeneous or a layered medium, the proposed acceleration techniques reveals to be not effective when applied to the resulting infinite double integrals of oscillating and slowly decaying functions.
As an application of this inequality, we construct new starlike function of order [beta] which can be expressed in terms of double integrals of some suitable function in the class H.
We introduce the following terms for simplifying the double integrals and equations.
Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, to appear in the Ramanujan Journal.
In this paper we extend the simplifying idea introduced in  from simple to double integrals. We will simplify the computation of the coefficients of the expansion without complicating the computation of the asymptotic sequence.
The original proof for this theorem, as in the case of de Bruijn's theorem, required the use of complicated mathematics involving double integrals and complex numbers.

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