Riemann sum

Let I=[a,b] be a closed intervalMathworldPlanetmath, f:I be boundedPlanetmathPlanetmathPlanetmath on I, n, and P={[x0,x1),[x1,x2),[xn-1,xn]} be a partition of I. The Riemann sumMathworldPlanetmath of f over I with respect to the partition P is defined as


where cj[xj-1,xj] is chosen arbitrary.

If cj=xj-1 for all j, then S is called a left Riemann sum.

If cj=xj for all j, then S is called a Riemann sum.

Equivalently, the Riemann sum can be defined as


where bj{f(x):x[xj-1,xj]} is chosen arbitrarily.

If bj=supx[xj-1,xj]f(x), then S is called an upper Riemann sum.

If bj=infx[xj-1,xj]f(x), then S is called a lower Riemann sum.

For some examples of Riemann sums, see the entry examples of estimating a Riemann integral.

Title Riemann sum
Canonical name RiemannSum
Date of creation 2013-03-22 11:49:17
Last modified on 2013-03-22 11:49:17
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 14
Author Wkbj79 (1863)
Entry type Definition
Classification msc 26A42
Related topic RiemannIntegral
Related topic RiemannStieltjesIntegral
Related topic LeftHandRule
Related topic RightHandRule
Related topic MidpointRule
Defines left Riemann sum
Defines right Riemann sum
Defines upper Riemann sum
Defines lower Riemann sum