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