Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

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)
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