Higher Engineering Mathematics, Sixth Edition

Some applications of integration 381

=

625

3

−

625

4

125

2

−

125

3

=

625

12

125

6

=

(

625

12

)(

6

125

)

=

5

2

=2.5

y=

1

2

∫ 5

0

y^2 dx

∫ 5

0

ydx

=

1

2

∫ 5

0

( 5 x−x^2 )^2 dx

∫ 5

0

( 5 x−x^2 )dx

=

1

2

∫ 5

0

( 25 x^2 − 10 x^3 +x^4 )dx

125

6

=

1

2

[

25 x^3

3

−

10 x^4

4

+

x^5

5

] 5

0

125

6

=

1

2

(

25 ( 125 )

3

−

6250

4

+ 625

)

125

6

=2.5

Hence the centroid of the area lies at (2.5, 2.5).

(Note from Fig. 38.10 that the curve is symmetrical
aboutx= 2 .5 and thusxcould have been determined
‘on sight’.)

Now try the following exercise

Exercise 150 Further problemson centroids

In Problems 1 and 2, find the position of the cen-

troidsof the areas bounded by the given curves, the

x-axis and the given ordinates.

1. y= 3 x+ 2 x= 0 ,x= 4 [(2.5, 4.75)]


2. y= 5 x^2 x= 1 ,x= 4 [(3.036, 24.36)]


3. Determine the position of the centroid of a
   sheet of metal formed by the curve
   y= 4 x−x^2 which lies above thex-axis.
   [(2, 1.6)]
   4. Findtheco-ordinatesofthecentroidofthearea
   which lies between the curvey/x=x−2and
   thex-axis. [(1,−0.4)]
   5. Sketch the curvey^2 = 9 xbetween the limits
   x=0andx=4. Determinethepositionofthe
   centroid of this area.
   [(2.4, 0)]

38.6 Theorem of Pappus

A theorem of Pappusstates:

‘If a plane area is rotated about an axis in its own plane

but not intersecting it, the volume of the solid formed is

given by the product of the area and the distance moved

by the centroid of the area’.

With reference to Fig. 38.11, when the curvey=f(x)

is rotated one revolution about the x-axis between

the limitsx=aandx=b,thevolumeV generated

is given by:

volumeV=(A)( 2 πy),from which,y=

V

2 πA

y

C

Area A

x 5 a 5 bxx

y 5 f(x)

y

Figure 38.11

Problem 9. (a) Calculate the area bounded by the

curvey= 2 x^2 ,thex-axis and ordinatesx=0and

x=3. (b) If this area is revolved (i) about the

x-axis and (ii) about they-axis, find the volumes of

the solids produced. (c) Locate the position of the

centroid using (i) integration, and (ii) the theorem

of Pappus.

(a) The required area is shown shaded in Fig. 38.12.

Area=

∫ 3

0

ydx=

∫ 3

0

2 x^2 dx

=

[

2 x^3

3

] 3

0

=18 square units