Main Page | See live article | Alphabetical index In mathematics, a Riemann sum is a method for approximating the values of integrals. Let it be supposed there is a function f: D → R where D, R &sube R and that there is a closed interval I = [a,b] such that I &sube D. If we have a finite set of points {x0, x1, x2, ... xn} such that a = x0 < x1 < x2 ... < xn = b, then this set creates a partition P = {[x0, x1), [x1, x2), ... [xn-1, xn]} of I. If is a partition with elements of , then the Riemann sum of over with the partition is defined as where xi-1 ≤ yi ≤ xi. The choice of yi is arbitrary. If yi = x_i-1 for all i, then S is called a left Riemann sum. If yi = xi, then S is called a right Riemann sum. Suppose we have where vi is the supremum of f over [xi-1, xi]; then S is defined to be an upper Riemann sum. Similarly, if vi is the infimum of f over [xi-1, xi], then S is a lower Riemann sum. See also: Bernhard Riemann Riemann integral Riemann-Stieltjes integral This article is from Wikipedia. All text is available under the terms of the GNU Free Documentation License.