Higher Engineering Mathematics, Sixth Edition

Irregular areas, volumes and mean values of waveforms 209

50

Graph of power/time

40

30

Power (kW)

20

10

0 123456

Time (hours)

7.0 21.5 42.0 49.5 37.0 10.0

Figure 19.10

Area under curve=(width of interval)

×(sum of mid-ordinates)

=( 1 )[7. 0 + 21. 5 + 42. 0

+ 49. 5 + 37. 0 + 10 .0]

=167kWh(i.e. a measure

of electrical energy)

(b) Average value of waveform

=

area under curve

length of base

=

167kWh

6h

=27.83kW

Alternatively, average value

=

sum of mid-ordinates

number of mid-ordinates

Problem 8. Fig. 19.11 shows a sinusoidal output

voltage of a full-wave rectifier. Determine, using

the mid-ordinate rule with 6 intervals, the mean

output voltage.

10

0 308608908 1808 3608 



2

2708

Voltage (V)

 3 

2

2 

Figure 19.11

Onecycleoftheoutputvoltageiscompletedinπradians

or 180◦. The base is divided into 6 intervals, each of

width 30◦. The mid-ordinate of each interval will lie at

15 ◦,45◦,75◦,etc.

At 15◦ the height of the mid-ordinate is

10sin15◦= 2 .588V.

At 45◦ the height of the mid-ordinate is

10sin45◦= 7 .071V, and so on.

The results are tabulated below:

Mid-ordinate Height of mid-ordinate

15 ◦ 10sin15◦= 2 .588V

45 ◦ 10sin45◦= 7 .071V

75 ◦ 10sin75◦= 9 .659V

105 ◦ 10sin105◦= 9 .659V

135 ◦ 10sin135◦= 7 .071V

165 ◦ 10sin165◦= 2 .588V

sum of mid-ordinates=38.636V

Mean or average value of output voltage

=

sum of mid-ordinates

number of mid-ordinates

=

38. 636

6

=6.439V

(With a larger number of intervals a more accurate

answer may be obtained.) For a sine wave the actual

mean value is 0.637×maximum value, which in this

problem gives 6.37V.

Problem 9. An indicator diagram for a steam

engine is shown in Fig. 19.12. The base line has

been divided into 6 equally spaced intervals and the

lengths of the 7 ordinates measured with the results

shown in centimetres. Determine (a) the area of the

indicator diagram using Simpson’s rule, and (b) the

mean pressure in the cylinder given that 1cm

represents 100kPa.

12.0 cm

3.6 4.0 3.5 2.9 2.2 1.7 1.6

Figure 19.12