Understanding Engineering Mathematics

0

3

2

,

x

y = x^2 − x

y^ =

2 x

−^ x

2

y

(1,0)

3

4

Figure 10.15Area betweeny= 2 x−x^2 andy=x^2 −x.

However, this will still be accommodated by integrating the difference

of the two functions over the range 0<x< 3 /2, as follows:

∫ 3 / 2

0

[2x−x^2 −(x^2 −x)]dx=

∫ 3 / 2

0

( 3 x− 2 x^2 )dx

=

9

8

units

10.2.6 Volume of a solid of revolution

➤

292 311➤

When the area under a curve between the limitsx=a,bis rotated once about thex-axis
asolid of revolutionis formed. For the purposes of illustration we will assume that the
curve is entirely above thex-axis, so that the area rotated is positive. See Figure 10.16.

x

y

y

y = f(x)

(^0) dx
ab
Figure 10.16Volume of revolution.