Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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