Mathematical Methods for Physics and Engineering : A Comprehensive Guide

2.2 INTEGRATION

a b

f(x)

x

Figure 2.7 An integral as the area under a curve.

2.2 Integration

The notion of an integral as the area under a curve will be familiar to the reader.

In figure 2.7, in which the solid line is a plot of a functionf(x), the shaded area

represents the quantity denoted by

I=

∫b

a

f(x)dx. (2.21)

This expression is known as thedefinite integraloff(x)betweenthelower limit

x=aand theupper limitx=b,andf(x) is called theintegrand.

2.2.1 Integration from first principles

The definition of an integral as the area under a curve is not a formal definition,

but one that can be readily visualised. The formal definition ofI involves

subdividing the finite intervala≤x≤binto a large number of subintervals, by

defining intermediate pointsξisuch thata=ξ 0 <ξ 1 <ξ 2 <···<ξn=b,and

then forming the sum

S=

∑n

i=1

f(xi)(ξi−ξi− 1 ), (2.22)

wherexiis an arbitrary point that lies in the rangeξi− 1 ≤xi≤ξi(see figure 2.8).

If nownis allowed to tend to infinity in any way whatsoever, subject only to the

restriction that the length of every subintervalξi− 1 toξitends to zero, thenS

might, or might not, tend to a unique limit,I. If it does then the definite integral

off(x) betweenaandbis defined as having the valueI. If no unique limit exists

the integral is undefined. For continuous functions and a finite intervala≤x≤b

the existence of a unique limit is assured and the integral is guaranteed to exist.