Calculus 3       Hour Exam 3   - Math 241    Spring, 2010

 

 

 

 

5.

 

 

 

 

 

Evaluate the integral  ∫   ∫  y dA  over the region  D  defined by  1  ≤  xy  ≤  4

                                     D

  

and  x  ≤  y  ≤  9x   using the substitution   xy  =  u2  and  y/x  =  v2 .  Sketch the

region in both xy-coordinates and  uv-coordinates.  Don’t forget the Jacobian!

 

Answer:  28/3                    

 

 

 

 

 

6.

 

 

Let the path   C  run straight from   (1, 1 )   to  ( √2 , √2 ), then counterclockwise around

the circle of radius  2  with center at  (0,0), stopping at  (-2, 0).  Evaluate

 

 

                      ∫    (y dx  -  x dy) / (x2  +  y2 )

 

Hint:  It is helpful to draw this path and identify each part contributing to  C.

 

Answer:   - 3π/ 4