composite trapezoidal rule

The composite trapezoidal rule is a method for approximating a definite integral by evaluating the integrand at n points. Let [a,b] be the interval of integration with a partition a=x0<x1<<xn=b. Then the formal rule is given by


The composite trapezoidal rule can also be applied to a partition which is uniformly spaced (i.e. ( xj-xj-1=h for all j{1,,n}). In this case, the formal rule is given by


Both expressions of the composite trapezoidal rule come from determining the areas of the figures in the corresponding graph. These figures are usually right trapezoidsMathworldPlanetmath, but may also be right trianglesMathworldPlanetmath or line segmentsMathworldPlanetmath on the x . See the entry on the trapezoidal rule for more details. See the sectionMathworldPlanetmath of the entry on examples of estimating a Riemann integral which deals with the composite trapezoidal rule for an illustration.

The composite trapezoidal rule uses the trapezoidal rule on each subinterval, which is readily observed from

abf(x)𝑑x = j=1nxj-1xjf(x)𝑑x

If f is Riemann integrable on [a,b], |f′′(x)|M for all x[a,b], and n is the number of intervalsMathworldPlanetmathPlanetmath of the partition used to approximate abf(x)𝑑x, then

Title composite trapezoidal rule
Canonical name CompositeTrapezoidalRule
Date of creation 2013-03-22 16:05:16
Last modified on 2013-03-22 16:05:16
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 11
Author Wkbj79 (1863)
Entry type Theorem
Classification msc 41A05
Classification msc 41A55
Synonym composite trapezoid rule
Related topic TrapezoidalRule