Example 6.20. Show that  [Graphics:Images/FunTheoremCalculusMod_gr_60.gif],  where C is the unit circle  [Graphics:Images/FunTheoremCalculusMod_gr_61.gif],  taken with positive orientation.

Figure 6.32  The simply connected domain D shown in Examples 6.19 and 6.20.

Explore Solution 6.20.

Enter the function  [Graphics:../Images/FunTheoremCalculusMod_gr_68.gif].

[Graphics:../Images/FunTheoremCalculusMod_gr_69.gif]





[Graphics:../Images/FunTheoremCalculusMod_gr_70.gif]

[Graphics:../Images/FunTheoremCalculusMod_gr_71.gif]

[Graphics:../Images/FunTheoremCalculusMod_gr_72.gif]

 

 


Another approach is to parameterize the integrand.

[Graphics:../Images/FunTheoremCalculusMod_gr_73.gif]





[Graphics:../Images/FunTheoremCalculusMod_gr_74.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2006 John H. Mathews, Russell W. Howell