integration

382 INTEGRAL CALCULUS
When area PQRS is rotated about axis XX the volume
generated is that of the pulley. The centroid of
the semicircular area removed is at a distance of 4r
3π
from its diameter (see ‘Engineering Mathematics 4th
edition’, page 471), i.e. 4(1.0) , i.e. 0.424 cm from
3π
PQ. Thus the distance of the centroid from XX is
5.0 − 0.424, i.e. 4.576 cm.
The distance moved through in one revolution by the
centroid is 2π(4.576) cm.
Area of semicircle = πr2
2 = π(1.0)2
2
By the theorem of Pappus,
= π 2 cm2
volume generated ( = area × distance moved by
π
)
centroid = (2π)(4.576).
2
i.e. volume of metal removed = 45.16 cm 3
Mass of metal removed = density × volume
= 8000 kg m −3 × 45.16
10 6 m3
= 0.3613 kg or 361.3g
volume of pulley = volume of cylindrical disc
− volume of metal removed
= π(5.0) 2 (2.0) − 45.16
= 111.9 cm 3
Mass of pulley = density × volume
= 8000 kg m −3 × 111.9
10 6 m3
= 0.8952 kg or 895.2 g
Now try the following exercise.
Exercise 152 Further problems on the theorem
of Pappus
1. A right angled isosceles triangle having a
hypotenuse of 8 cm is revolved one revolution
about one of its equal sides as axis. Determine
the volume of the solid generated using
Pappus’ theorem. [189.6 cm 3 ]
2. Using (a) the theorem of Pappus, and (b) integration,
determine the position of the centroid
of a metal template in the form of a quadrant
of a circle of radius 4 cm. (The equation of a
circle, centre 0, radius r is x 2 + y 2 = r 2 ).
⎡
⎤
On the centre line, distance
2.40 cm from the centre, ⎥
i.e. at co-ordinates ⎦
(1.70, 1.70)
⎢
⎣
3.(a) Determine the area bounded by the curve
y = 5x 2 , the x-axis and the ordinates
x = 0 and x = 3.
(b) If this area is revolved 360 ◦ about (i) the
x-axis, and (ii) the y-axis, find the volumes
of the solids of revolution produced
in each case.
(c) Determine the co-ordinates of the centroid
of the area using (i) integral calculus,
and (ii) the theorem of Pappus.
⎡
⎤
(a) 45 square units
⎢(b) (i) 1215π cubic units ⎥
⎣ (ii) 202.5π cubic units⎦
(c) (2.25, 13.5)
4. A metal disc has a radius of 7.0 cm and is
of thickness 2.5 cm. A semicircular groove of
diameter 2.0 cm is machined centrally around
the rim to form a pulley. Determine the volume
of metal removed using Pappus’theorem
and express this as a percentage of the original
volume of the disc. Find also the mass of
metal removed if the density of the metal is
7800 kg m −3 .
[64.90 cm 3 , 16.86%, 506.2 g]
For more on areas, mean and r.m.s. values, volumes
and centroids, see ‘Engineering Mathematics 4th
edition’, Chapters 54 to 57.
38.7 Second moments of area of
regular sections
The first moment of area about a fixed axis of a
lamina of area A, perpendicular distance y from the
centroid of the lamina is defined as Ay cubic units.
The second moment of area of the same lamina as
above is given by Ay 2 , i.e. the perpendicular distance