Higher Engineering Mathematics, Sixth Edition

Numerical integration 439

Now try the following exercise

Exercise 174 Further problemson the

mid-ordinate rule

In Problems 1 to 4, evaluate the definite integrals

using themid-ordinate rule, giving the answers

correct to 3 decimal places.

1.

∫ 2

0

3

1 +t^2

dt (Use 8 intervals) [3.323]

2.

∫ π

2

0

1

1 +sinθ

dθ (Use 6 intervals) [0.997]

3.

∫ 3

1

lnx

x

dx (Use 10 intervals) [0.605]

4.

∫ π

3

0

√

(cos^3 x)dx (Use 6 intervals) [0.799]

45.4 Simpson’srule

The approximation made with the trapezoidal rule is to
join the top of two successive ordinates by a straight
line, i.e. by using a linear approximation of the form
a+bx. With Simpson’s rule, the approximation made
is to join the tops of three successive ordinates by a
parabola, i.e. by using a quadratic approximation of the
forma+bx+cx^2.
Figure 45.3 shows a parabolay=a+bx+cx^2 with
ordinates y 1 ,y 2 and y 3 at x=−d,x=0andx=d
respectively.
Thus the width of each of the two intervals isd.The
area enclosed by the parabola, thex-axis and ordinates
x=−dandx=dis given by:
∫d

−d

(a+bx+cx^2 )dx=

[

ax+

bx^2

2

+

cx^3

3

]d

−d

=

(

ad+

bd^2

2

+

cd^3

3

)

−

(

−ad+

bd^2

2

−

cd^3

3

)

= 2 ad+

2

3

cd^3 or

1

3

d( 6 a+ 2 cd^2 ) (3)

y

y 1 y 2 y 3

y  abxcx^2

d O d x

Figure 45.3

Since y=a+bx+cx^2 ,

at x=−d,y 1 =a−bd+cd^2

at x= 0 ,y 2 =a

and at x=d,y 3 =a+bd+cd^2

Hence y 1 +y 3 = 2 a+ 2 cd^2

And y 1 + 4 y 2 +y 3 = 6 a+ 2 cd^2 (4)

Thus the area under the parabola between x=−d

and x=d in Fig. 45.3 may be expressed as

1

3 d(y^1 +^4 y^2 +y^3 ), from equations (3) and (4), and the

result is seen to be independent of the position of the

origin.

Let a definite integral be denoted by

∫b

aydx and

represented by the area under the graph ofy=f(x)

betweenthelimitsx=aandx=b,asshowninFig.45.4.

The range of integration,b−a, is divided into aneven

number of intervals, say 2n, each of widthd.

Since an even number of intervals is specified, an odd

number of ordinates, 2n+1, exists. Let an approxima-

tiontothe curve over the first twointervalsbe a parabola

of the formy=a+bx+cx^2 which passes through the

tops of the three ordinatesy 1 ,y 2 andy 3. Similarly, let

an approximation to the curve over the next two inter-

vals be the parabola which passes through the tops of

the ordinatesy 3 ,y 4 andy 5 , and so on.

Then

∫b

a

ydx

≈

1

3

d(y 1 + 4 y 2 +y 3 )+

1

3

d(y 3 + 4 y 4 +y 5 )

+

1

3

d(y 2 n− 1 + 4 y 2 n+y 2 n+ 1 )