Riemann Integral

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Riemann integral

[′rē‚män ‚int·ə·grəl]
The Riemann integral of a real function ƒ(x) on an interval (a,b) is the unique limit (when it exists) of the sum of ƒ(ai )(xi -xi-1), i = 1, …, n, taken over all partitions of (a,b), a = x0<>a1<>x1< ⋯=""><>an <>xn = b, as the maximum distance between xi and xi-1tends to zero.
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 It might be outdated or ideologically biased.

Riemann Integral


the usual definite integral. The definition of the Riemann integral was in essence given by A. Cauchy in 1823. Cauchy, however, applied his definition to continuous functions. In a paper written in 1853 and published in 1867, B. Riemann was the first to point out the necessary and sufficient conditions for the existence of the definite integral. In modern terms, these conditions can be stated as follows: for the definite integral of a function to exist over some interval, it is necessary and sufficient that (1) the interval be finite, (2) the function to be bounded on the interval, and (3) the set of discontinuities of the function on the interval have Lebesgue measure zero (seeMEASURE OF A SET).

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
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Xiao starts by describing sets, relations, functions, cardinals, ordinals, reals, basic theorems and sequence limits, proceeding to Riemann integrals, Riemann-Stieltjes integrals, Lebesque-Radon-Stieltjes integrals, metric spaces, continuous maps, normed linear spaces, Banach spaces via operators and functionals, and Hilbert spaces and their operators.