Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

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.
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