Exercise 3.  Consider the integral  ,  where C is the positively oriented upper semicircle of radius 1, centered at 0.  

3 (a).  Give a Riemann sum approximation for the integral by selecting    and using the

points     and   .  

Solution 3 (a).

See text and/or instructor's solution manual.

Answer.  The approximation simplifies to  .

Solution.  Use the following:  ,  ,  ,  and

                                

Make the substitutions and get  

                              

                                 

This solution is complements of the authors.

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