Introduction to Chapter 3

Equation ‎3.1. The Simpson's rule. The minimum number of observations required to apply Simpson's rule is three. The first and last observation are entered in the first set of parentheses, regardless of being odd observations (i.e. the 1^st and 3^rd). At its minimum, the second set of parentheses is empty, since the first and the last observation go into the first set, and 2^nd observation being even goes into the last set. Therefore, a minimum of five observations are needed before the second set of parentheses is present in the calculation. The Simpson's rule is sometimes called the 'one-third' rule since the method uses a weighted average of the values at the endpoints and the midpoint of each parabolic segment, hence the 1⁄3 factor.

\[A_{\mathrm{total}} \approx \frac{h}{3} \times \left[\underbrace{(y_0\;+\;y_n)}_\text{first and last observations}+2\underbrace{(y_2+\ldots+y_{n-2})}_\text{odd-number observations}+4\underbrace{(y_1+\ldots+y_{n-1})}_\text{even-number observations}\right]\]

where \(h = \frac{x_n-x_0}{n}\) is the distance between each observation and \(n\) is the number of subintervals \((n≥2; \text{even})\)