Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

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

(

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)
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