Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

PRELIMINARY CALCULUS


2.2.10 Infinite and improper integrals

The definition of an integral given previously does not allow for cases in which


either of the limits of integration is infinite (aninfinite integral)orforcases


in whichf(x) is infinite in some part of the range (animproper integral), e.g.


f(x)=(2−x)−^1 /^4 near the point x= 2. Nevertheless, modification of the


definition of an integral gives infinite and improper integrals each a meaning.


In the case of an integralI=

∫b
af(x)dx, the infinite integral, in whichbtends
to∞, is defined by


I=

∫∞

a

f(x)dx= lim
b→∞

∫b

a

f(x)dx= lim
b→∞

F(b)−F(a).

As previously,F(x) is the indefinite integral off(x) and limb→∞F(b)meansthe


limit (or value) thatF(b) approaches asb→∞; it is evaluatedaftercalculating


the integral. The formal concept of a limit will be introduced in chapter 4.


Evaluate the integral

I=

∫∞


0

x
(x^2 +a^2 )^2

dx.

Integrating, we findF(x)=−^12 (x^2 +a^2 )−^1 +cand so


I= lim
b→∞

[


− 1


2(b^2 +a^2 )

]



(


− 1


2 a^2

)


=


1


2 a^2

.


For the case of improper integrals, we adopt the approach of excluding the

unbounded range from the integral. For example, if the integrandf(x) is infinite


atx=c(say),a≤c≤bthen


∫b

a

f(x)dx= lim
δ→ 0

∫c−δ

a

f(x)dx+ lim
→ 0

∫b

c+

f(x)dx.

Evaluate the integralI=

∫ 2


0 (2−x)

− 1 / (^4) dx.
Integrating directly,
I= lim→ 0


[


−^43 (2−x)^3 /^4

] 2 −


0 = lim→ 0

[


−^43 ^3 /^4


]


+^4323 /^4 =


( 4


3

)


23 /^4 .


2.2.11 Integration in plane polar coordinates

In plane polar coordinatesρ, φ, a curve is defined by its distanceρfrom the


origin as a function of the angleφbetween the line joining a point on the curve


to the origin and thex-axis, i.e.ρ=ρ(φ). The area of an element is given by

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