Iterated Integral

iterated integral

[′īd·ə‚rād·əd ′int·ə·grəl]
An integral over an area or volume designated to be performed by successive integrals over line segments.
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 It might be outdated or ideologically biased.

Iterated Integral


a concept of the integral calculus. Let

I = ∫∫Sf(x, y) dxdy

be the double integral of the function f(x, y) over the region S bounded by the lines x = a and x = b and the curves y = φ1(x) and y = φ2(x). If certain conditions on f(x, y), φ1(x), and φ2(x) are fulfilled, the double integral can be calculated by the formula

where x is kept constant when the inner integral is calculated. The calculation of a double integral thus reduces to two calculations of ordinary integrals, or to the calculation of what is called an iterated integral.

Geometrically, the reduction of a double integral to an iterated integral means that the volume of a cylindroid can be calculated both by dividing it into elementary columns and by dividing it into elementary layers parallel to the yz-plane. The order of integration in the iterated integral can be changed—that is, integration may be performed first with respect to x and then with respect to y—if certain conditions are imposed on f(x, y) and S. The iterated integral is defined in a similar manner for functions of more than two variables.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
we define the n-fold iterated integral of f with respect to [??] by
The first approach, Riemann-Liouville approach, is to iterate the integral with respect to certain weight function and replace the iterated integral by single integral through Leibniz-Cauchy formula and then replace the factorial function by the Gamma function.
Then, the function [K.sub.n+1](t) is the n +1 iterated integral of the function [K.sub.0](t), that may be written in the following way:
The integration limits for the inner iterated integral are [y.sub.1](x) and [y.sub.2] (x) in each case.
Choosing [x.sub.2] and [x.sub.d-1] as the outer variables of integration, we can compute this integral as the iterated integral
In 1979 Rab [4] (see also [5]) has given the explicit upper bound on the following useful iterated integral inequality