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128 chapter 6evaluate the function at the two endpoints and in the middle (Table 6.1):∫ xi+hx i−h∫ xi+h ∫ xif(x) dx = f(x) dx + f(x) dxx i x i−h≃ h 3 f i−1 + 4h 3 f i + h 3 f i+1. (6.16)Simpson’s rule requires the elementary integration to be over pairs of intervals,which in turn requires that the total number of intervals be even or that the numberof points N be odd. In order to apply Simpson’s rule to the entire interval, we addup the contributions from each pair of subintervals, counting all but the first andlast endpoints twice:∫ baf(x)dx ≃ h 3 f 1 + 4h 3 f 2 + 2h 3 f 3 + 4h 3 f 4 + ···+ 4h 3 f N−1 + h 3 f N. (6.17)In terms of our standard integration rule (6.3), we have{ hw i =3 , 4h 3 , 2h 3 , 4h 3 , ..., 4h 3 , h }3(Simpson’s rule). (6.18)The sum of these weights provides a useful check on your integration:N∑w i =(N − 1)h. (6.19)i=1Remember, the number of points N must be odd for Simpson’s rule.6.2.3 Integration Error (Analytic Assessment)In general, you should choose an integration rule that gives an accurate answerusing the least number of integration points. We obtain a crude estimate of theapproximation or algorithmic error E and the relative error ɛ by expanding f(x) ina Taylor series around the midpoint of the integration interval. We then multiplythat error by the number of intervals N to estimate the error for the entire region[a, b]. For the trapezoid and Simpson rules this yields( ) [b − a]3E t = ON 2 f (2) ,( ) [b − a]5E s = ON 4 f (4) ,ɛ t,s = E t,sf . (6.20)−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 128