Exercises for Section 6.4.  The Fundamental Theorems of Integration

     For Exercises 1-14, find the value of the definite integral using Theorem 6.9 and explain why you are justified in using it.

Exercise 1.  ,  where C is the line segment from  .   
Solution 1.

Exercise 2.  ,  where C is the line segment from  .
Solution 2.

Exercise 3.  ,  where C is the line segment from  .  
Solution 3.

Exercise 4.  ,  where C is the line segment from  .   
Solution 4.

Exercise 5.  ,  where C is the line segment  .  
Solution 5.

Exercise 6.  ,  where C is the line segment from  .    
Solution 6.

Exercise 7.  ,  where C is the line segment from  .  
Solution 7.

Exercise 8.  ,  where C is the line segment from  .  
Solution 8.

Exercise 9.  ,  where C is the line segment from  .  
Solution 9.

Exercise 10.  ,  where C is the line segment from  .  
Solution 10.

Exercise 11.  ,  where C is the line segment from  .  
Solution 11.

Exercise 12.  ,  where C is the line segment from  .  
Solution 12.

Exercise 13.  ,  where C is the line segment from  .  
Solution 13.

Exercise 14.  ,  where C is the line segment from  .  
Solution 14.

Exercise 15.  Show that  ,  where C is the line segment from  ,  by parametrizing C.  
Solution 15.

Exercise 16.  Let    be points in the right half-plane and let C be the line segment joining them.

Show that  .  
Solution 16.

Exercise 17.  Let    be the principal branch of the square root function.  

17 (a).  Evaluate  ,  where C is the line segment joining  .  
Solution 17 (a).

17 (b).  Evaluate  ,  where C is the right half of the circle    joining  .  
Solution 17 (b).

Exercise 18.  Using partial fraction decomposition, show that if z lies in the right half-plane
and C is the line segment joining  , then  

    .  
Solution 18.

Exercise 19.  Let    be analytic for all z and let C be any contour joining the points  .  
Show that  

        .  
Solution 19.

Exercise 20.  Compare the various methods for evaluating contour integrals.  What are the limitations of each method?  
Solution 20.

Exercise 21.  Explain how the fundamental theorem of calculus studied in complex analysis and
the fundamental theorem of calculus studied in calculus are different.  
How are they similar?  
Solution 21.

Exercise 22.  Show that  ,  where C is the upper half of  .    
Solution 22.

Exercise 23.  Consider the following computation for evaluating ,  
where C is the line segment from  .  

.  

Is this a valid computation ?  Why ?  Justify your answer.   
Solution 23.

(c) 2008 John H. Mathews, Russell W. Howell