Higher Engineering Mathematics

H

Integral calculus

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
termednumerical 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 methods 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 a required definite integral be denoted by

∫b
aydx
and be represented by the area under the graph of

y

y = f(x)

y 1 y 2 y 3 y 4 yn+ 1

O x = ax = bx

ddd

Figure 45.1

y=f(x) between the limitsx=aandx=bas shown

in 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 labelledy 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 ordi-

nates by straight lines. Each interval is thus a trapez-

ium, and since the area of a trapezium is given by:

area=

1

2

(sum of parallel sides) (perpendicular

distance between them) then

∫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)

Problem 1. (a) Use integration to evaluate, cor-

rect to 3 decimal places,

∫ 3

1

2

√

x

dx(b) Use the

trapezoidal rule with 4 intervals to evaluate the

integral in part (a), correct to 3 decimal places.