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.

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).

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In a calculus textbook (see [1] or [2]), the Riemann integral [[integral].
Using examples and very useful illustrations she describes the basic building blocks of real analysis, including sets and set notations, functions, and sequences, the real numbers, measuring distances, sets and limits, continuity, real-valued functions, completeness, compactness, connectedness, differentiation of functions of one real variable, iteration and the contraction mapping theorem, the Riemann integral, sequences of functions, differentiation of functions, truth and probability, number properties, exponents, sequences in R, limits of functions from R to R, doubly indexed sequences, sub-sequences and convergence, series of real numbers, probing the definition of the Reimann integral, power series, and Newton's method.