Chapter 19

Irregular areas, volumes and

mean values of waveforms

19.1 Areas of irregular figures

Areas of irregular plane surfaces may be approximately
determined by using(a) a planimeter,(b) thetrapezoidal
rule, (c) the mid-ordinate rule, and (d) Simpson’s rule.
Such methods may be used, for example, by engineers
estimatingareas ofindicatordiagrams ofsteam engines,
surveyors estimating areas of plots of land or naval
architects estimating areas of water planes or transverse
sections of ships.

(a) Aplanimeteris an instrument for directly mea-

suring small areas bounded by an irregular curve.

(b) Trapezoidal rule

To determine the areasPQRSin Fig. 19.1:

Q R

P S

ddd

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

ddd

Figure 19.1

(i) DividebasePSintoanynumberofequalinte-

rvals,eachofwidthd(thegreaterthenumber

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. 19.2:

B C

A D

ddd

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

ddd

Figure 19.2

(i) Divide baseADinto any number of equal

intervals, each of widthd(the greater the

number of intervals,thegreater theaccuracy).

(ii) Erect ordinates in the middle of each interval

(shown by broken lines in Fig. 19.2).