Simpson’s one-third rule

Numerical integration of a variable / measured at a set of equally spaced values of the independent variable x. The integral 7(1,7) = [ydx is approximated with both the trapezoidal rule (a) and Simpson s one-third rule b). In each case, the value of Y is given by the area under the heavy lines. The light lines in b) represent extensions of the three parabolic sections that are used to construct this approximation. 

Simpson s Rules. There are better procedures for approximating the integral of a function that make use of quadratic and cubic forms rather than linear segments. The simplest of these is Simpson s one-third rule ... 

In cases where n is divisible neither by 2 nor by 3, the range of integration may be split into two parts, one for Simpson s one-third rule, the other for Simpson s three-eighths rule. Alternatively, if the curve is approximately linear in one or two intervals, the trapezoidal rule may be used in these intervals. 

The errors resulting from the use of Simpson s one-third rule are much smaller than those associated with the use of the trapezoidal rule. This is illustrated by the example given in Fig. 1. The points shown in this figure were generated from the function j = 10 - + 0.03/ and are given below ... 

Simpson s one-third rule (three-point) (Figure A-3). A more accurate evaluation of the integral can be found with the application of Simpson s rule ... 

Simpson s three-eighths rule (four-point) (Figure A-4). An improved version of Simpson s one-third rule can be made by applying Simpson s second rule ... 

This version of Simpson s rule is sometimes called Simpson s one-third rule because of the 3 in the denominator. There is another version, called Simpson s five-eighths rule, which corresponds to fitting third-degree polynomials to four points at a time. 

If h represents the common distance of the ordinates apart, we ba e the familiar result known as Simpson s one-third rule, thus,... 

Examples.—(1) Compare Simpson s one-third rule and the three-eighths rule when h = 1, with the result of the integration of... 

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