Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PRELIMINARY CALCULUS

∆x

∆s ∆y

y=f(x)

x

f(x)

Figure 2.11 The distance moved along a curve, ∆s, corresponding to the

small changes ∆xand ∆y.

In plane polar coordinates,

ds=

√

(dr)^2 +(rdφ)^2 ⇒ s=

∫r 2

r 1

√

1+r^2

(

dφ

dr

) 2

dr.

(2.43)

Find the length of the curvey=x^3 /^2 fromx=0tox=2.

Using (2.42) and noting thatdy/dx=^32

√

x, the lengthsof the curve is given by

s=

∫ 2

0

√

1+^94 xdx

=

[

2

3

( 4

9

)(

1+^94 x

) 3 / 2 ] 2

0

= 278

[(

1+^94 x

) 3 / 2 ] 2

0

= 278

[(

11

2

) 3 / 2

− 1

]

.

Surfaces of revolution

Consider the surfaceSformed by rotating the curvey=f(x) about thex-axis

(see figure 2.12). The surface area of the ‘collar’ formed by rotating an element

of the curve,ds, about thex-axis is 2πy ds, and hence the total surface area is

S=

∫b

a

2 πy ds.

Since (ds)^2 =(dx)^2 +(dy)^2 from (2.41), the total surface area between the planes

x=aandx=bis

S=

∫b

a

2 πy

√

1+

(

dy

dx

) 2

dx. (2.44)