Basic Engineering Mathematics, Fifth Edition

260 Basic Engineering Mathematics

16400m^3 = 16400 × 106 cm^3.

Since 1 litre=1000cm^3 ,

capacity of reservoir=

16400 × 106

1000

litres

= 16400000 = 16. 4 × 106 litres

Now try the following Practice Exercise

PracticeExercise 111 Volumes of irregular

solids (answers on page 352)

1. The areas of equidistantly spaced sections of
   the underwater form of a small boat are as fol-
   lows:
   1.76, 2.78, 3.10, 3.12, 2.61, 1.24 and 0.85m^2.
   Determine the underwater volume if the
   sections are 3m apart.


2. To estimate the amount of earth to be
   removed when constructing a cutting, the
   cross-sectional area at intervals of 8m were
   estimated as follows:
   0, 2.8, 3.7, 4.5, 4.1, 2.6 and 0m^3.
   Estimate the volume of earth to be excavated.


3. Thecircumferenceofa 12mlonglogoftimber
   of varying circular cross-section is measured
   at intervals of 2m along its length and the
   results are as follows. Estimate the volume of
   the timber in cubic metres.

Distance from

one end(m) 0 2 4 6

Circumference(m) 2.80 3.253.94 4.32

Distance from

one end(m) 8 10 12

Circumference(m) 5.16 5.82 6.36

28.3 Mean or average values of waveforms

The mean or average value,y,of the waveform shown

in Figure 28.6 is given by

y=

area under curve

length of base,b

If the mid-ordinaterule is used to find the area under the

curve, then

y=

sum of mid-ordinates

number of mid-ordinates

(

=

y 1 +y 2 +y 3 +y 4 +y 5 +y 6 +y 7

7

for Figure 28.6

)

y 1 y 2 y 3 y 4 y 5 y 6 y 7

dddd

b

ddd

y

Figure 28.6

For asine wave, the mean or average value

(a) over one complete cycle is zero (see

Figure 28.7(a)),

(b) overhalfacycleis 0. 637 ×maximum valueor

2

π

×maximum value,

(c) of a full-wave rectified waveform (see

Figure 28.7(b)) is 0. 637 ×maximum value,

(d) of a half-wave rectified waveform (see

Figure 28.7(c)) is 0. 318 ×maximum value

or

1

π

×maximum value.

V

(^0) t
Vm
V
0
(a) (b)
t
Vm
(c)
V
(^0) t
Vm
Figure 28.7