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

3 (b).  Compute the integral exactly by selecting a parametrization for C and applying Theorem 6.1.

Solution 3 (b).

See text and/or instructor's solution manual.

Answer.  .  

Solution Method I.  A parametrization for C is      for   .

                    The contour      for   .

The function is    and the curve is    and we obtain

                                 and  ,  

Use these substitutions and get  


                                 

The last two real integrals are computed using

                                 
                    
                               and

                                 

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

Solution Method II.  We could use a method that uses the following complex computations.

Solution.  A parametrization for C is      for   .

          The contour      for   .

The function is    and the curve is    for  .   Then we obtain

                       and   ,  
                    
                    then

                       

Here we have used the calculations indicated by equation (6-8) in Section 6.1:

                        

We are really really done.   

Aside.  We can let Mathematica double check our work.

We are really really really done.   

Remark concerning parts (a) and (b).  The approximation in part (a) was   

and the exact value in part (b) is   

We are really really really  really done.   

Aside.  After we have developed the Cauchy-Goursat Theorem in Section 6.3 and the Fundamental Theorem of Calculus in Section 6.4

we will be able to compute the integral in an easy fashion:

                    .  

Aside.  We can let Mathematica double check our work.

This solution is complements of the authors.

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