Mathematical Methods for Physics and Engineering : A Comprehensive Guide

27.4 NUMERICAL INTEGRATION

xi xi+1/ 2 xi+1 xi xi+1 xi− 1 xi xi+1

h hhh

f(x) f(x)

(a) (b) (c)

fi

fi

fi+1

fi+1 fi+1

fi− 1

Figure 27.4 (a) Definition of nomenclature. (b) The approximation in using

the trapezium rule;f(x) is indicated by the broken curve. (c) Simpson’s rule

approximation;f(x) is indicated by the broken curve. The solid curve is part

of the approximating parabola.

numerical evaluations ofIare based on regardingIas the area under the curve

off(x) between the limitsx=aandx=band attempting to estimate that area.

The simplest methods of doing this involve dividing up the intervala≤x≤b

intoNequal sections, each of lengthh=(b−a)/N. The dividing points are

labelledxi, withx 0 =a,xN=b,irunning from 0 toN. The pointxiis a distance

ihfroma. The central value ofxin a strip (x=xi+h/2) is denoted for brevity

byxi+1/ 2 , and for the same reasonf(xi) is written asfi. This nomenclature is

indicated graphically in figure 27.4(a).

So that we may compare later estimates of the area under the curve with the

true value, we next obtain an exact expression forI, even though we cannot

evaluate it. To do this we need to consider only one strip, say that betweenxi

andxi+1. For this strip the area is, using Taylor’s expansion,

∫h/ 2

−h/ 2

f(xi+1/ 2 +y)dy=

∫h/ 2

−h/ 2

∑∞

n=0

f(n)(xi+1/ 2 )

yn

n!

dy

=

∑∞

n=0

f(in+1)/ 2

∫h/ 2

−h/ 2

yn

n!

dy

=

∑∞

neven

fi(+1n)/ 2

2

(n+1)!

(

h

2

)n+1

. (27.35)

It should be noted that, in this exact expression, only the even derivatives of

fsurvive the integration and all derivatives are evaluated atxi+1/ 2. Clearly