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In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann . One very common application is in numerical integration , i.e., approximating the area of functions or lines on a graph, where it is also known as the rectangle rule .
One popular restriction is the use of "left-hand" and "right-hand" Riemann sums. In a left-hand Riemann sum, t i = x i for all i, and in a right-hand Riemann sum, t i = x i + 1 for all i. Alone this restriction does not impose a problem: we can refine any partition in a way that makes it a left-hand or right-hand sum by subdividing it at each t i.
Discrete integral calculus is the study of the definitions, properties, and applications of the Riemann sums. The process of finding the value of a sum is called integration. In technical language, integral calculus studies a certain linear operator.
A partition of an interval being used in a Riemann sum. The partition itself is shown in grey at the bottom, with the norm of the partition indicated in red. In mathematics, a partition of an interval [a, b] on the real line is a finite sequence x 0, x 1, x 2, …, x n of real numbers such that a = x 0 < x 1 < x 2 < … < x n = b.
When the chosen tags give the maximum (respectively, minimum) value of each interval, the Riemann sum is known as the upper (respectively, lower) Darboux sum. A function is Darboux integrable if the upper and lower Darboux sums can be made to be arbitrarily close to each other for a sufficiently small mesh. Although this definition gives the ...
Riemann's original use of the explicit formula was to give an exact formula for the number of primes less than a given number. To do this, take F(log(y)) to be y 1/2 /log(y) for 0 ≤ y ≤ x and 0 elsewhere. Then the main term of the sum on the right is the number of primes less than x.
Bernhard Riemann at the Mathematics Genealogy Project; The Mathematical Papers of Georg Friedrich Bernhard Riemann; Riemann's publications at emis.de; O'Connor, John J.; Robertson, Edmund F., "Bernhard Riemann", MacTutor History of Mathematics Archive, University of St Andrews; Bernhard Riemann – one of the most important mathematicians
The Rectangle method and Riemann sum are the same; I propose that they should be merged. Klbrain 20:55, 26 April 2016 (UTC) The Riemann sum is a more technical and complicated way of explaining integrals. I do believe, however, that Rectangle method should be merged instead with Trapezoidal rule. Both these quadrature methods are very similar ...