Chapter 45

Numerical integration

45.1 Introduction

Even with advanced methods of integration there are
many mathematical functions which cannot be inte-
grated by analytical methods and thus approximate
methods have then to be used. Approximate methods of
definite integrals may be determined by what is termed
numerical integration.
It may be shown that determining the value of a definite
integral is, in fact, finding the area between a curve, the
horizontal axis and the specified ordinates. Three meth-
ods of finding approximate areas under curves are the
trapezoidal rule, the mid-ordinate rule and Simpson’s
rule, and these rules are used as a basis for numerical
integration.

45.2 The trapezoidal rule

Let arequireddefiniteintegral bedenotedby

∫b
aydxand
be represented by the area under the graph ofy=f(x)
between thelimitsx=aandx=bas shownin Fig. 45.1.
Let the range of integration be divided intonequal
intervals each of widthd, such thatnd=b−a,i.e.

d=

b−a
n
The ordinates are labelled y 1 ,y 2 ,y 3 ,...,yn+ 1 as
shown.
An approximation to the area under the curve may be
determined by joining the tops of the ordinates by
straight lines. Each interval is thus a trapezium, and
since the area of a trapezium is given by:

area=

1

2

(sum of parallel sides) (perpendicular

distance between them) then

y 1 y 2 y 3 y 4

xa

yn 1

O xb

yf(x)

x

y

ddd

Figure 45.1

∫b

a

ydx≈

1

2

(y 1 +y 2 )d+

1

2

(y 2 +y 3 )d

+

1

2

(y 3 +y 4 )d+···

1

2

(yn+yn+ 1 )d

≈d

[

1

2

y 1 +y 2 +y 3 +y 4 +···+yn

+

1

2

yn+ 1

]

i.e.the trapezoidal rule states:

∫b

a

ydx≈

(

width of

interval

){

1

2

(

first+last

ordinate

)

+

(

sum of remaining

ordinates

)} (1)