Numerical Integration: How the Error of the Trapezoidal and Simpson Rules Falls with the Number of Intervals

Abstract

Many integrals that arise in engineering cannot be evaluated by a formula, and are instead computed numerically, by dividing the region under the curve into strips and adding up their areas; how accurately this is done depends on how many strips are used and, on the rule, used to estimate each strip's area, and the two common rules, the trapezoidal and Simpson's, differ greatly in how fast their error falls as more strips are used. This study examines numerical integration and how the error of the trapezoidal and Simpson rules falls with the number of intervals. The error of each rule, in approximating a known integral, was examined as the number of intervals was increased, so that how fast the error fell, and how the two rules differed, could be seen. The error of both rules fell as the number of intervals grew, but at very different rates: the trapezoidal rule, which approximates each strip by a straight-topped trapezium, had an error that fell with the square of the number of intervals, so doubling the intervals cut the error to about a quarter; Simpson's rule, which fits a curve through each pair of strips, had an error that fell with the fourth power of thenumber of intervals, so doubling the intervals cut the error to about a sixteenth. Simpson's rule was therefore far more accurate than the trapezoidal for the same number of intervals, and its advantage widened rapidly as the intervals increased, reaching a given accuracy with far fewer intervals. There was a practical floor, however, set by rounding in the arithmetic, below which using more intervals ceased to help and could even harm. The higher-order rule is preferred because it buys accuracy far more cheaply, which matters when each evaluation of the function is costly. The study shows that numerical integration error falls with the number of intervals, faster for higher-order rules, and it discusses the implications for computing integrals in engineering.  KEYWORDS: Numerical integration, Quadrature, Trapezoidal rule, Simpson's rule, Order of accuracy, Error, Step size, Numerical methods, engineering mathematics