# Trapezoidal Rule

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## trapezoidal rule

[¦trap·ə¦zȯid·əl ′rül]*a*to

*b*of a real function ƒ(

*x*) is approximated by where

*x*

_{0}=

*a*,

*x*

_{j }=

*x*

_{j-1}+ (

*b - a*)/

*n*for

*j*= 1, 2, …,

*n*- 1.

## Trapezoidal Rule

(or trapezoid rule), a formula for the approximate evaluation of definite integrals. It has the form

where *f _{m}* =

*f*(

*a*+

*mh*),

*h*= (

*b*–

*a*)/

*n*, and

*m*= 0, 1, . . . .,

*n*.

The use of the trapezoidal rule may be understood in geometric terms by regarding the definite integral *I* as expressing the area under the curve *y* = *f*(*x*) from *x* = a to *x* = *b*—that is, the area of the region bounded by the segment on the *x*-axis between the points *a* and *b*, the perpendiculars to the *x*-axis at these points (the lengths of the perpendiculars are given by the ordinates *f*_{0} and *f _{n}*), and the graph of

*f*(

*x*). In applying the trapezoidal rule, we replace this area by the sum of the areas of the trapezoids the lengths of whose bases are given by the pairs of ordinates

*f*,

_{m}*f*

_{m + 1}(

*m*= 0, 1, . . .,

*n*– 1).

The error resulting from the use of the trapezoidal rule is

where *a* ≤ ξ ≤ *b*. Formulas of greater accuracy for the approxi mate evaluation of definite integrals are discussed in APPROXIMATE INTEGRATION.