Mathematical Methods for Physics and Engineering : A Comprehensive Guide

2.2 INTEGRATION

S

y

b

V

f(x)

dx

a x

ds

Figure 2.12 The surface and volume of revolution for the curvey=f(x).

Find the surface area of a cone formed by rotating about thex-axis the liney=2x

betweenx=0andx=h.

Using (2.44), the surface area is given by

S=

∫h

0

(2π)2x

√

1+

[

d

dx

(2x)

] 2

dx

=

∫h

0

4 πx

(

1+2^2

) 1 / 2

dx=

∫h

0

4

√

5 πx dx

=

[

2

√

5 πx^2

]h

0

=2

√

5 π(h^2 −0) = 2

√

5 πh^2 .

We note that a surface of revolution may also be formed by rotating a line

about they-axis. In this case the surface area betweeny=aandy=bis

S=

∫b

a

2 πx

√

1+

(

dx

dy

) 2

dy. (2.45)

Volumes of revolution

The volumeVenclosed by rotating the curvey=f(x) about thex-axis can also

be found (see figure 2.12). The volume of the disc betweenxandx+dxis given

bydV=πy^2 dx.Hence the total volume betweenx=aandx=bis

V=

∫b

a

πy^2 dx. (2.46)