Mathematical Methods for Physics and Engineering : A Comprehensive Guide

2.2 INTEGRATION

m

f(x)

a b x

Figure 2.10 The mean valuemof a function.

Find the mean valuemof the functionf(x)=x^2 between the limitsx=2andx=4.

Using (2.40),

m=

1

4 − 2

∫ 4

2

x^2 dx=

1

2

[

x^3

3

] 4

2

=

1

2

(

43

3

−

23

3

)

=

28

3

.

Finding the length of a curve

Finding the area between a curve and certain straight lines provides one example

of the use of integration. Another is in finding the length of a curve. If a curve

is defined byy=f(x) then the distance along the curve, ∆s, that corresponds to

small changes ∆xand ∆yinxandyis given by

∆s≈

√

(∆x)^2 +(∆y)^2 ; (2.41)

this follows directly from Pythagoras’ theorem (see figure 2.11). Dividing (2.41)

through by ∆xand letting ∆x→0weobtain§

ds

dx

=

√

1+

(

dy

dx

) 2

.

Clearly the total lengthsof the curve between the pointsx=aandx=bis then

given by integrating both sides of the equation:

s=

∫b

a

√

1+

(

dy

dx

) 2

dx. (2.42)

§Instead of considering small changes ∆xand ∆yand letting these tend to zero, we could have

derived (2.41) by considering infinitesimal changesdxanddyfrom the start. After writing (ds)^2 =

(dx)^2 +(dy)^2 , (2.41) may be deduced by using the formal device of dividing through bydx. Although

not mathematically rigorous, this method is often used and generally leads to the correct result.