Higher Engineering Mathematics, Sixth Edition

Irregular areas, volumes and mean values of waveforms 205

Problem 2. A river is 15m wide. Soundings of

the depth are made at equal intervals of 3m across

the river and are as shown below.

Depth (m) 0 2.2 3.3 4.5 4.2 2.4 0

Calculate the cross-sectional area of the flow of

water at this point using Simpson’s rule.

From para. (d) above,

Area=^13 ( 3 )[( 0 + 0 )+ 4 ( 2. 2 + 4. 5 + 2. 4 )

+ 2 ( 3. 3 + 4. 2 )]

=( 1 )[0+ 36. 4 +15]=51.4m^2

Now try the following exercise

Exercise 82 Further problemson areas of

irregular figures

1. Plot a graph ofy= 3 x−x^2 by completing
   a table of values ofyfromx=0tox=3.
   Determine the area enclosed by the curve, the
   x-axis and ordinatex=0andx=3by(a)the
   trapezoidal rule, (b) the mid-ordinate rule and
   (c) by Simpson’s rule. [4.5square units]


2. Plot the graph ofy= 2 x^2 +3 betweenx= 0
   andx=4. Estimate the area enclosed by the
   curve, the ordinatesx=0andx=4, and the
   x-axis by an approximate method.
   [54.7square units]


3. The velocity of a car at one second intervals is
   given in the following table:

timet(s) 0 1 2 3 4 5 6

velocity

v(m/s)

0 2.0 4.5 8.0 14.0 21.0 29.0

Determine the distance travelled in 6seconds

(i.e. the area under the v/t graph) using

Simpson’s rule. [63.33m]

4. The shape of a piece of land is shown in
   Fig. 19.4. To estimate the area of the land,
   a surveyor takes measurements at intervals
   of 50m, perpendicular to the straight portion
   with the results shown (the dimensions being
   in metres). Estimate the area of the land in
   hectares (1ha= 104 m^2 ). [4.70ha]

50 50 50 50

140 160 200 190 180 130

50 50

Figure 19.4

5. The deck of a ship is 35m long. At equal
   intervals of 5m the width is given by the
   following table:

Width (m) 0 2.8 5.2 6.5 5.8 4.1 3.0 2.3

Estimate the area of the deck. [143m^2 ]

19.2 Volumes of irregular solids

If the cross-sectional areasA 1 ,A 2 ,A 3 ,...of an irregular

solid bounded by two parallel planes are known at equal

intervals of widthd(as shown in Fig. 19.5), then by

Simpson’s rule:

volume,V=

d

3

[(A 1 +A 7 )+4(A 2 +A 4

+A 6 )+2(A 3 +A 5 )]

A 1 A 2 A 3 A 4 A 5 A 6 A 7

dd d d d d d

Figure 19.5

Problem 3. A tree trunk is 12m in length and has

a varying cross-section. The cross-sectional areas at

intervals of 2m measured from one end are:

0.52, 0.55, 0.59, 0.63, 0.72, 0.84, 0.97m^2

Estimate the volume of the tree trunk.