The Chemistry Maths Book, Second Edition

276 Chapter 9Functions of several variables

Line integrals are important in several branches of the physical sciences. For example,

when generalized to curves in three dimensions, they provide a method for repre-

senting and calculating the work done by a force along an arbitrary path in ordinary

space; we return to this topic in Chapter 16.

The line integral (9.48) can be converted into an ordinary integral over either

variable when the equation of the curve C is known. Thus, given the curvey 1 = 1 f(x),

replacement ofdyin (9.48) by gives

(9.49)

EXAMPLE 9.23Find the value of the line integral (9.48) whenF 1 = 1 −y, G 1 = 1 xy, and

C is the line in Figure 9.8 from A to B (x 1 = 11 tox 1 = 10 ).

The equation of the line isy 1 = 111 − 1 x. Then dy 1 = 1 −dx, and, by

equation (9.49),

0 Exercises 57, 58

In general, the value of a line integral depends on the path of integration between the

end points. This is demonstrated in the following example.

EXAMPLE 9.24Find the value of the line integral (9.48) when Fand Gare as in

Example 9.23, but C is now the circular arc shown in Figure 9.9.

The equation of the circular arc is Then

and

=−−−













Z =+

1

0

22

1

4

1

3

xxdx

π

I=−+y dx xy dy













Z

C

dy

x

x

dx

x

y

= dx

−

−

=−

1

2

yx=+ 1 −

2

.

=− =Z

0

1

2

1

2

3

()xdx

ZZ

C

−+













=−−−−













ydx xydy x x x dx

1

0

()() 11

IFxyGxy

dy

dx

dx

a

b

=,+,

















Z () ()

dy

dx

dx

0 1

1

• 
  
  
  
  • 
    
  
  
  
  

a

b

c

x

y

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Figure 9.8

0 1

1

• 
  
  
  
  • 
    
  
  
  
  

a

b

c

x

y

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Figure 9.9