Higher Engineering Mathematics, Sixth Edition

210 Higher Engineering Mathematics

(a) The width of each interval is

12. 0

6

cm. Using

Simpson’s rule,

area=^13 ( 2. 0 )[( 3. 6 + 1. 6 )+ 4 ( 4. 0

+ 2. 9 + 1. 7 )+ 2 ( 3. 5 + 2. 2 )]

=^23 [5. 2 + 34. 4 + 11 .4]

=34cm^2

(b) Mean height of ordinates

=

area of diagram

length of base

=

34

12

= 2 .83cm

Since 1cm represents 100kPa, the mean pressure

in the cylinder

= 2 .83cm×100kPa/cm=283kPa.

Now try the following exercise

Exercise 84 Further problems on mean or

average values of waveforms

1. Determine the mean value of the periodic
   waveforms shown in Fig. 19.13 over a half
   cycle. [(a) 2A (b) 50V (c) 2.5A]

(^01020)
22
2
Current (A)
t (ms)
0510
2100
100
Voltage (V)
t (ms)
(a)
(b)
Figure 19.13
015 30
25
5
Current (A)
t (ms)
(c)
Figure 19.13 (Continued)

2. Find the average value of the periodic wave-
   forms shown in Fig. 19.14 over one complete
   cycle. [(a) 2.5V (b) 3A]

(b)

0

5

Current (A)

(a)

(^0246810)
10
Voltage (mV)
t (ms)
246810 t (ms)
Figure 19.14

3. 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 0

Plot 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.

[0.093As, 3.1A]

4. Determine, using an approximate method, the
   average value of a sine wave of maximum
   value 50V for (a) a half cycle and (b) a
   complete cycle. [(a) 31.83V (b) 0]