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 ?
Structural reliability could be expressed as multiple integrals as follows:
They assume students have completed a first-year calculus-based physics course, along with a good first-year course in calculus; some of the problems also require knowledge of multi-variable calculus, especially multiple integrals. Students should have access to mathematical computer programs and be able to use them; they particularly recommend software that can deal with both linear and non-linear differential equations.
All multiple integrals converge absolutely and we can change the order of the integrations.
Therefore, if the performance analysis problem at hand involves high-dimension multiple integrals, Gaussian quadrature will become too inaccurate to be used as a tool for numerical computation, and if that is the case, we have to go back to rely on Monte-Carlo simulations for the results we need.
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.
The multi-time cost functional can be introduced either using a path independent curvilinear integral, or using a multiple integral. Using a similar argument as in [11], page 144, variational problems with multiple integrals can be converted to problems with curvilinear integrals and conversely.
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.
Throughout this work a knowledge and understanding of time scales and time-scale notation is assumed; for an excellent introduction to the calculus on time scales, see Bohner and Peterson [5,6] for a general overview, the paper introducing nabla derivatives by Atici and Guseinov [3], and the introduction of multiple integrals on time scales by Bohner and Guseinov [4].
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.
Tseng, On certain multiple integral inequalities related to Hermite-Hadamard inequality, Utilitas Mathematica 62(2002), 131-142.

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