Irregular areas and volumes, and mean values 263
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
(b) the mean pressure in the cylinder given that
1cm represents 100kPa.12.0 cm3.6 4.0 3.5 2.9 2.2 1.7 1.6Figure 28.12(a) The width of each interval is12. 0
6cm.Using
Simpson’s rule,area=1
3( 2. 0 )[( 3. 6 + 1. 6 )+ 4 ( 4. 0 + 2. 9 + 1. 7 )+ 2 ( 3. 5 + 2. 2 )]=2
3[5. 2 + 34. 4 + 11 .4]=34cm^2(b) Mean height of ordinates=
area of diagram
length of base=34
12= 2 .83cmSince 1cm represents 100kPa,
mean pressure in the cylinder
= 2 .83cm×100kPa/cm=283kPaNow try the following Practice Exercise
PracticeExercise 112 Mean or average
values of waveforms (answers on page 352)- Determine the mean value of the periodic
waveforms shown in Figure 28.13 over a half
cycle. - Find the average value of the periodic wave-
forms shown in Figure 28.14 over one com-
plete cycle.
(a)Current (A) 0210 20
22t(ms)(b)Voltage (V) 01005102100t(ms)(c)Current (A) 0515 30
25t(ms)Figure 28.13Voltage (mV)
010(^246810) t(ms)
Current (A)
0
5
(^246810) t(ms)
Figure 28.14
- Analternatingcurrent hasthefollowingvalues
at equal intervals of 5ms:
Time (ms) 0 5 10 15 20 25 30
Current (A) 0 0.9 2.6 4.9 5.8 3.5 0Plot a graph of current against time and esti-
mate the area under the curve over the 30ms
period, using the mid-ordinate rule, and deter-
mine its mean value.- Determine, using an approximate method, the
average value of a sine wave of maximum
value 50V for (a) a half cycle (b) a complete
cycle. - An indicator diagram of a steam engine is
12cm long. Seven evenly spaced ordinates,
including the end ordinates, are measured as
follows:
5.90, 5.52, 4.22, 3.63, 3.32, 3.24 and 3.16cm.
Determine the area of the diagram and the
mean pressure in the cylinder if 1cm repre-
sents 90kPa.