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

Solution 11.

See text and/or instructor's solution manual.

Answer.   .  

                    The curve  ,  for  .

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

                        and   ,  
                     
                    then

                       

The last two real integrals are computed using

                        
                    
                    
                    and
                    

                      

We are 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:

                      

We are really done.   

Aside.  We can let Mathematica double check our work.

This solution is complements of the authors.

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