Concise Physical Chemistry

c03 JWBS043-Rogers September 13, 2010 11:24 Printer Name: Yet to Come

52 THE THERMODYNAMICS OF SIMPLE SYSTEMS

PROBLEMS AND EXAMPLE

Example 3.1 Line Integrals

What is the line integral of the function f(x,y)=xyover the parabolic curve

y=f(x)=x^2 /2from(x,y)=( 0 , 0 )to

(

1 ,^12

)

?

Solution 3.1 One way of writing a line integral of the functionI=

∫

Cf(x,y)ds

over the curve C is to specifyy=f(x) andds=

(

1 +

(

dy

dx

) 2 )^1 /^2

dx. For example,

integrating the function f(x,y)=xyover the parabolic curvey=f(x)=x^2 / 2

from(x,y)=( 0 , 0 )to

(

1 ,^12

)

. We have (Steiner, 1996)

f(x,y)=xy=x

(

x^2

2

)

=

x^3

2

and

dy

dx

=

d

(

x^2

2

)

dx

=x

Thus,

I=

∫

C

f(x,f(x))

(

1 +

(

dy

dx

) 2 ) 1 / 2

dx=

∫ 1

0

x^3

2

(

1 +x^2

)^1

(^2) dx

=

1

2

∫ 1

0

(

x^6 +x^8

) 1 / 2

dx

where the limits of integration are the limits onx. Integration by Mathcad©Cgives

1

2

∫ 1

0

(

x^6 +x^8

). 5

dx= 0. 161

Problem 3.1

One expression of a line integral is

∫

C

F(x,y)dx+G(x,y)dy

where the subscriptCindicates a line (or curve) integral. IfF(x,y)=−y,G(x,y)=

xy, and the line is the diagonal fromx=1toy=1 (Fig. 3.8). Carry out the

integration.