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trapezoidal rule[¦trap·ə¦zȯid·əl ′rül]
(or trapezoid rule), a formula for the approximate evaluation of definite integrals. It has the form
where fm = 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 f0 and fn), 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 fm, fm + 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.