integration

380 INTEGRAL CALCULUS
625
=
3 − 625 625
4
125
2 − 125 =
12
125
3 6
( )( )
625 6
=
= 5 12 125 2 = 2.5
y =
=
1
2
1
2
∫ 5
0
∫ 5
0
∫ 5
0
y 2 1
dx
2
=
y dx
∫ 5
0
∫ 5
0
(5x − x 2 ) 2 dx
(5x − x 2 )dx
(25x 2 − 10x 3 + x 4 )dx
125
6
[
] 5
1 25x 3
− 10x4 + x5
2 3 4 5
0
=
125
6
(
1 25(125)
− 6250 )
+ 625
2 3 4
=
= 2.5
125
6
Hence the centroid of the area lies at (2.5, 2.5).
(Note from Fig. 38.10 that the curve is symmetrical
about x = 2.5 and thus x could have been determined
‘on sight’.)
Now try the following exercise.
Exercise 151 Further problems on
centroids
In Problems 1 and 2, find the position of the centroids
of the areas bounded by the given curves,
the x-axis and the given ordinates.
1. y = 3x + 2 x = 0, x = 4 [(2.5, 4.75)]
2. y = 5x 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 = 4x − x 2 which lies above the x-axis.
[(2, 1.6)]
4. Find the co-ordinates of the centroid of the
area which lies between the curve y/x = x−2
and the x-axis.
[(1, −0.4)]
5. Sketch the curve y 2 = 9x between the limits
x = 0 and x = 4. Determine the position of
the centroid of this area.
[(2.4, 0)]
38.6 Theorem of Pappus
A theorem of Pappus states:
‘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 curve y = f (x)
is rotated one revolution about the x-axis between
the limits x = a and x = b, the volume V generated
is given by:
volume V = (A)(2πy), from which, y =
V
2πA
Figure 38.11
Problem 9. (a) Calculate the area bounded by
the curve y = 2x 2 , the x-axis and ordinates x = 0
and x = 3. (b) If this area is revolved (i) about the
x-axis and (ii) about the y-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
[ 2x
3
=
3
y dx =
] 3
0
∫ 3
0
2x 2 dx
= 18 square units