Higher Engineering Mathematics, Sixth Edition

Some applications of integration 383

When areaPQRSis rotated about axisXXthe vol-
ume generated is that of the pulley. The centroid of the

semicircular area removed is at a distance of

4 r

3 π

from its

diameter (see ‘Engineering Mathematics6th edition’,

Chapter 58), i.e.

4 ( 1. 0 )

3 π

,i.e.0.424cmfromPQ. Thus

the distance of the centroid fromXXis 5. 0 − 0 .424,
i.e. 4.576cm.
The distance moved through in one revolution by the
centroid is 2π( 4. 576 )cm.

Area of semicircle=

πr^2

2

=

π( 1. 0 )^2

2

=

π

2

cm^2

By the theorem of Pappus,
volume generated=area×distance moved by

centroid=

(π

2

)

( 2 π)( 4. 576 ).

i.e.volume of metal removed=45.16cm^3

Massofmetal removed=density×volume

=8000kgm−^3 ×

45. 16

106

m^3

= 0 .3613kg or 361.3g

volume of pulley=volume of cylindrical disc

−volume of metal removed

=π( 5. 0 )^2 ( 2. 0 )− 45. 16

=111.9cm^3

Mass of pulley=density×volume

=8000kg m−^3 ×

111. 9

106

m^3

=0.8952kgor895.2g

Now try the following exercise

Exercise 151 Further problems on the

theorem of Pappus

1. A right angled isosceles triangle having a
   hypotenuse of 8cm is revolved one revolution
   about one of its equal sides as axis. Deter-
   mine the volume of the solid generated using
   Pappus’ theorem. [189.6cm^3 ]


2. Using (a) the theorem of Pappus, and (b) inte-
   gration, determine the position of the centroid
   of a metal template in the form of a quadrant

of a circle of radius 4cm. (The equation of a

circle, centre 0, radiusrisx^2 +y^2 =r^2 ).

⎡

⎢

⎢

⎢

⎣

On the centre line, distance

2 .40cm from the centre,

i.e. at co-ordinates

( 1. 70 , 1. 70 )

⎤

⎥

⎥

⎥

⎦

3. (a) Determine the area boundedby thecurve
   y= 5 x^2 ,thex-axis and the ordinates
   x=0andx=3.
   (b) If this area is revolved 360◦about (i) the
   x-axis, and (ii) they-axis, find the vol-
   umesofthesolidsofrevolutionproduced
   in each case.
   (c) Determine the co-ordinates of the cen-
   troid of the area using (i) integral calcu-
   lus, and (ii) the theorem of Pappus.
   ⎡
   ⎢
   ⎢
   ⎢
   ⎣

(a) 45 square units

(b) (i) 1215πcubic units

(ii) 202.5πcubic units

(c) (2.25, 13.5)

⎤

⎥

⎥

⎥

⎦

4. A metal disc has a radius of 7.0cm and is
   of thickness 2.5cm. A semicircular groove of
   diameter 2.0cm is machined centrally around
   the rim to form a pulley. Determine the vol-
   ume of metal removed using Pappus’ theorem
   and express this as a percentage of the origi-
   nal volume of the disc. Find also the mass of
   metal removed if the density of the metal is
   7800kgm−^3.
   [64.90cm^3 , 16.86%, 506.2g]

For more on areas, mean and r.m.s. values, volumes and

centroids, see ‘Engineering Mathematics6th edition’,

Chapters 55 to 58.

38.7 Second moments of area of

regular sections

Thefirst moment of areaabout a fixed axis of a lamina

of areaA, perpendicular distanceyfrom the centroid

of the lamina is defined asAycubic units. Thesecond

moment of areaof the same lamina as above is given

byAy^2 , i.e. the perpendicular distance from the centroid

of the area to the fixed axis is squared.