Higher Engineering Mathematics

Graphs

20

Irregular areas, volumes and mean

values of waveforms

20.1 Areas of irregular figures

Areas of irregular plane surfaces may be approxi-

mately determined by using (a) a planimeter, (b) the

trapezoidal rule, (c) the mid-ordinate rule, and (d)

Simpson’s rule. Such methods may be used, for

example, by engineers estimating areas of indicator

diagrams of steam engines, surveyors estimating

areas of plots of land or naval architects estimating

areas of water planes or transverse sections of ships.

(a)A planimeteris an instrument for directly mea-

suring small areas bounded by an irregular

curve.

(b)Trapezoidal rule

To determine the areasPQRSin Fig. 20.1:

Figure 20.1

(i) Divide basePSinto any number of equal

intervals, each of width d (the greater

the number of intervals, the greater the

accuracy).

(ii) Accurately measure ordinatesy 1 ,y 2 ,y 3 , etc.

(iii) AreasPQRS

=d

[

y 1 +y 7

2

+y 2 +y 3 +y 4 +y 5 +y 6

]

In general, the trapezoidal rule states:

Area=

(

width of

interval

)[

1

2

(

first+

last

ordinate

)

+

sum of

remaining

ordinates

]

(c)Mid-ordinate rule

To determine the areaABCDof Fig. 20.2:

Figure 20.2

(i) Divide baseADinto any number of equal

intervals, each of width d (the greater

the number of intervals, the greater the

accuracy).

(ii) Erect ordinates in the middle of each interval

(shown by broken lines in Fig. 20.2).

(iii) Accurately measure ordinatesy 1 ,y 2 ,y 3 , etc.

(iv) AreaABCD=d(y 1 +y 2 +y 3 +y 4 +y 5 +y 6 ).

In general, the mid-ordinate rule states:

Area=

(

width of

interval

)(

sum of

mid-ordinates

)