Multiple Integral

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multiple integral

[′məl·tə·pəl ′int·ə·grəl]
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 ?
In the remaining three weeks, the topic of double integrals was taught using the flipped classroom approach.
Thus, students were expected to come to the lesson prepared with Khan Academy materials and under the guidance of the researcher students had more opportunity to focus on and discuss the topic of double integrals.
On the base of the abovementioned, it is possible to express the double integrals in (B.
For simplification, the coordinate transformation by changing of variables in the double integral in (B.
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 [3] from simple to double integrals.
Thus, anyone who can follow a little bit of calculus/algebra/statistics can follow the main body of the book without difficult as long as one is not deterred by a few double integrals and some not so obvious properties of a Poisson process.
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.
Quraishi, Evaluation of Certain Elliptic Type Single, Double Integrals of Ramanujan and Erdelyi, J.
Dragomir, An Ostrowski Type Inequality for Double Integrals and Applications for Cubature Formulae, RGMIA Res.