trapezoidal rule

The trapezoidal ruleMathworldPlanetmath is a method for approximating a definite integral by evaluating the integrand at two points. The formal rule is given by


where h=b-a.

This rule comes from determining the area of a right trapezoidMathworldPlanetmath with bases ( of lengths f(a) and f(b) respectively and a height ( of length h. When using a graph to illustrate the trapezoidal rule, the height of the right trapezoid is actually horizontal and the bases are vertical. This may be confusing to someone who is seeing the trapezoidal rule for the first time. An example is shown below.


The figure in red need not be a right trapezoid. If either f(a)=0 or f(b)=0, the figure will be a right triangleMathworldPlanetmath. If both f(a)=0 and f(b)=0, the figure will be a line segmentMathworldPlanetmath. In any case, the same rule for approximating the corresponding definite integral is used.

The trapezoidal rule is the first Newton-Cotes quadrature formula. It has degree of precision 1. This means it is exact for polynomials of degree ( less than or equal to one. We can see this with a example.

If f is Riemann integrablePlanetmathPlanetmath on [a,b] with |f′′(x)|M for all x[a,b], then


Following is an example of the trapezoidal rule.

Using the fundamental theorem of calculusMathworldPlanetmathPlanetmath shows


In this case, the trapezoidal rule gives the exact value,


It is important to note that most calculus books give the wrong definition of the trapezoidal rule. Typically, they define it to be what is actually the composite trapezoidal rule, which uses the trapezoidal rule on a specified number of subintervals. Some examples of calculus books that define the trapezoidal rule to be what is actually the composite trapezoidal rule are:

  • Stewart, James. Calculus. Pacific Groves, : International Thomson Publishing Co., 1995.

  • Bittinger, Marvin L. Calculus. Reading, : Addison-Wesley Publishing Co., 1989.

Also note the trapezoidal rule can be derived by integrating a linear interpolation or by using the method of undetermined coefficients. The latter is probably a bit easier.

Title trapezoidal rule
Canonical name TrapezoidalRule
Date of creation 2013-03-22 13:39:43
Last modified on 2013-03-22 13:39:43
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 25
Author Wkbj79 (1863)
Entry type Definition
Classification msc 41A55
Classification msc 41A05
Synonym trapezoid rule
Synonym trapezium rule
Related topic CompositeTrapezoidalRule