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 It might be outdated or ideologically biased.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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The topics include Kyiv from the fall of 1943 through 1946: the rebirth of mathematics, two consequences of extension of local maps of Banach spaces: applications and examples, Hasse-Schmidt derivations and the Cayley-Hamilton theorem for exterior algebras, some binomial formulae for non-commuting operators, and the complete metric space of Riemann integrable functions and differential calculus in it.
One says that f(t) is fuzzy Riemann integrable in [??] [member of] [R.sub.F] if, for any [epsilon] >0, there exists [delta] > 0 such that for any division P = {[u, v]; [xi]} with the norms [DELTA](P) < [delta] one has
f is called Riemann integrable on [a, b] if there exist some I in S with the following property: for any positive numbers [epsilon] and [lambda] with [lambda] < 1 there is a positive number [delta] ([epsilon], [lambda]) such that
Let f be a continuous function from [a, b] to S such that [V.sub.t[member of][a,b] [parallel]f (t) [parallel] [member of] [L.sup.0.sub.+], then f is Riemann integrable in the ([epsilon], [lambda]) topology on [a, b].
We say that f' is Riemann integrable over [a, b] if
It is easy to see that if f' is bounded in a closed interval and satisfying 11, then f is Riemann integrable.
can not be made as small as possible for some 1 [less than or equal to] i [less than or equal to] n, then f' is not Riemann integrable over [a, b], see Example 13 when considering f(x) = [square root of x] over [0,1].
Hence, if f is an elementary function over an interval [a, b], it is clear that the total error [absolute value of f(b) - f(a) - [[summation].sup.n.sub.i=1] f'([x.sub.i]) [h.sub.i]] is small, f' is Riemann integrable and we have [[integral].sup.b.sub.a] f' = f(b) - f(a).
provided u is L-Lipschitzian and f is Riemann integrable and with the property that there exists the constants m, M [member of] R such that
However, we learn later that the family of Riemann integrable functions-denoted by R(f), is only a subset of Lebesgue integrable functions-denoted by L(f) and the family of Henstock-Kurzweil integrable functions-denoted by HK(f )-is an extension for Lebesgue integrable functions.
Theorem 6 If f is Riemann integrable over [c, b] for each c [member of] (a, b] and improper Riemann integrable over the interval [a, b].
Since f is improper Riemann integrable on [a, b], we have