midpoint rule

The midpoint rule for computing the Riemann integral abf(x)𝑑x is


If the Riemann integral is considered as a measureMathworldPlanetmath of area under a curve, then the expressions f(a+(j-12)(b-an)) the of the rectanglesMathworldPlanetmath, and b-an is the common of the rectangles.

The Riemann integral can be approximated by using a definite value for n rather than taking a limit. In this case, the partition is {[a,a+b-an),,[a+(b-a)(n-1)n,b]}, and the function is evaluated at the midpointsMathworldPlanetmathPlanetmathPlanetmath of each of these intervalsMathworldPlanetmathPlanetmath. Note that this is a special case of a Riemann sumMathworldPlanetmath in which the xj’s are evenly spaced and the cj’s chosen are the midpoints.

If f is Riemann integrable on [a,b] such that |f′′(x)|M for every x[a,b], then

Title midpoint rule
Canonical name MidpointRule
Date of creation 2013-03-22 15:57:44
Last modified on 2013-03-22 15:57:44
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 16
Author Wkbj79 (1863)
Entry type Theorem
Classification msc 41-01
Classification msc 28-00
Classification msc 26A42
Related topic LeftHandRule
Related topic RightHandRule
Related topic RiemannSum
Related topic ExampleOfEstimatingARiemannIntegral