Some applications of integration 379

Problem 6. Determine the area enclosed by the

two curvesy=x^2 andy^2 = 8 x.Ifthisareais

rotated 360◦about thex-axis determine the volume

of the solid of revolution produced.

At the points of intersection the co-ordinates of the
curves are equal. Sincey=x^2 theny^2 =x^4. Hence
equating they^2 values at the points of intersection:

x^4 = 8 x

from which, x^4 − 8 x= 0

and x(x^3 − 8 )= 0

Hence, at the points of intersection,x=0andx=2.
Whenx=0,y=0andwhenx=2,y=4. The points
of intersection of the curvesy=x^2 and y^2 = 8 xare
therefore at (0,0) and (2,4). A sketch is shown in
Fig. 38.8. Ify^2 = 8 xtheny=

√

8 x.

Shaded area

=

∫ 2

0

(√

8 x−x^2

)

dx=

∫ 2

0

(√

8

)

x

1

(^2) −x^2
)
dx

⎡
⎣
(√
8
)x^32
3
2
−
x^3
3
⎤
⎦
2
0

{√
8
√
8
3
2
−
8
3
}
−{ 0 }

16
3
−
8
3

8
3
= 2
2
3
square units
y
01 x
2
4
2
y 5 x^2
y^258 x
(or y 5 ŒŒ(8x)
Figure 38.8
The volume produced by revolving the shaded area
about thex-axis is given by:
{(volume produced by revolvingy^2 = 8 x)
−(volume produced by revolvingy=x^2 )}
i.e.volume=
∫ 2
0
π( 8 x)dx−
∫ 2
0
π(x^4 )dx
=π
∫ 2
0
( 8 x−x^4 )dx=π
[
8 x^2
2
−
x^5
5
] 2
0
=π
[(
16 −
32
5
)
−( 0 )
]
=9.6πcubic units
Now try the following exercise
Exercise 149 Further problems on volumes

1. The curvexy=3 is revolved one revolution
   about thex-axis between the limitsx=2and
   x=3. Determine the volume of the solid
   produced. [1.5πcubic units]


2. The area between
   y
   x^2

=1andy+x^2 =8is

rotated 360◦about thex-axis. Find the vol-

ume produced. [170^23 πcubic units]

3. The curvey= 2 x^2 +3 is rotated about (a) the
   x-axis between the limitsx=0andx=3,
   and (b) they-axis, between the same limits.
   Determine the volume generated in each case.
   [(a) 329.4π(b) 81π]


4. The profile of a rotor blade is bounded by the
   linesx= 0. 2 ,y= 2 x,y=e−x,x=1andthe
   x-axis. The blade thicknesstvaries linearly
   withxand is given by:t=( 1. 1 −x)K, where
   K is a constant.
   (a) Sketchtherotorblade,labellingthelimits.
   (b) Determine, using an iterative method, the
   value ofx, correct to 3 decimal places,
   where 2x=e−x
   (c) Calculate the cross-sectional area of the
   blade, correct to 3 decimal places.
   (d) Calculatethevolumeof thebladein terms
   of K, correct to 3 decimal places.
   [(b) 0.352 (c) 0.419 square units
   (d) 0.222K]