Chapter 28

Irregular areas and volumes,

and mean values

28.1 Areas of irregular figures

Areas of irregular plane surfaces may be approximately
determined by using

(a) a planimeter,

(b) the trapezoidal rule,

(c) the mid-ordinate rule, or

(d) Simpson’s rule.

Such methods may be used by, for example, 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.

There are many different kinds of planimeters but

all operate in a similar way. A pointer on the

planimeter is used to trace around the boundary

of the shape. This induces a movement in another

part of the instrument and a reading of this is used

to establish the area of the shape.

(b) Trapezoidal rule
To determine the areaPQRSin Figure 28.1,

(i) Divide basePSinto any number of equal

intervals, each of widthd (the greater the

number of intervals, the greater the accu-

racy).

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

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

Q R

P

dddddd

S

Figure 28.1

(iii) Area PQRS

=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

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

B

A

dddddd

D

Figure 28.2

To determine the areaABCDof Figure 28.2,

DOI: 10.1016/B978-1-85617-697-2.00028-4