082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

17 Numericalintegration
Contents 17.1 Introduction 496
17.2 Trapeziumrule 496
17.3 Simpson’srule 500
Reviewexercises17 505
17.1 INTRODUCTION
Many functions, for example sinx 2 and ex
, cannot be integrated analytically. Integrationofsuchfunctionsmustbeperformednumerically.Thissectionoutlinestwosimple
x
numericaltechniques--thetrapeziumruleandSimpson’srule.Moresophisticatedones
exist and there are many excellent software packages available which implement these
methods.
17.2 TRAPEZIUMRULE
We wish to find the area undery(x), fromx = a tox = b, that is we wish to evaluate
∫ b
ydx. The required area is divided intonstrips, each of widthh. Note that the width,
a
h, of each strip is given byh = b −a . Each strip is then approximated by a trapezium.
n
AtypicaltrapeziumisshowninFigure17.1.Theareaofthetrapeziumis 1 2 h[y i +y i+1 ].
Summing the areas of all the trapezia will yield an approximation tothe total area:
area oftrapezia = h 2 (y 0 +y 1 )+h 2 (y 1 +y 2 )+···+h 2 (y n−1 +y n )
= h 2 (y 0 +2y 1 +2y 2 +2y 3 +···+2y n−1 +y n )