Higher Engineering Mathematics, Sixth Edition

204 Higher Engineering Mathematics

(iii) Accurately measure ordinatesy 1 ,y 2 ,y 3 ,etc.

(iv) AreaABCD=d(y 1 +y 2 +y 3 +y 4 +y 5 +y 6 )

In general, the mid-ordinate rule states:

Area=

(

width of

interval

)(

sum of

mid-ordinates

)

(d) Simpson’s rule

To determine the areaPQRSof Fig. 19.1:

(i) Divide basePSintoanevennumber of inter-

vals, each of widthd(the greater the number

of intervals, the greater the accuracy).

(ii) Accurately measure ordinatesy 1 ,y 2 ,y 3 ,etc.

(iii) AreaPQRS=

d

3

[(y 1 +y 7 )+ 4 (y 2 +y 4 +

y 6 )+ 2 (y 3 +y 5 )]

In general, Simpson’s rule states:

Area=

1

3

(

width of

interval

)[(

first+last

ordinate

)

+ 4

(

sum of even

ordinates

)

+ 2

(

sum of remaining

odd ordinates

)]

Problem 1. A car starts from rest and its speed is

measured every second for 6s:

Timet(s) 01 2 3 4 5 6

Speedv(m/s) 0 2.5 5.5 8.75 12.517.524.0

Determine the distance travelled in 6 seconds (i.e.

the area under thev/tgraph), by (a) the trapezoidal

rule, (b) the mid-ordinate rule, and (c) Simpson’s

rule.

A graph of speed/time is shown in Fig. 19.3.

(a) Trapezoidal rule(see para. (b) above)

The time base is divided into 6 strips each of

width1s,andthelengthoftheordinatesmeasured.

30

25

Graph of speed/time

20

15

Speed (m/s)

10

5

0123456

Time (seconds)

1.252.54.05.57.08.7510.7512.515.017.520.2524.0

Figure 19.3

Thus

area=( 1 )

[(

0 + 24. 0

2

)

+ 2. 5 + 5. 5

+ 8. 75 + 12. 5 + 17. 5

]

=58.75m

(b) Mid-ordinate rule(see para. (c) above)

The time base is divided into 6 strips each of width

1second.

Mid-ordinatesare erected as shownin Fig. 19.3by

the broken lines. The length of each mid-ordinate

is measured. Thus

area=( 1 )[1. 25 + 4. 0 + 7. 0 + 10. 75

+ 15. 0 + 20 .25]

=58.25m

(c) Simpson’s rule(see para. (d) above)

The time base is divided into 6 strips each of

width1s,andthelengthoftheordinatesmeasured.

Thus

area=^13 ( 1 )[( 0 + 24. 0 )+ 4 ( 2. 5 + 8. 75

+ 17. 5 )+ 2 ( 5. 5 + 12. 5 )]

=58.33m