Formulas 

1. What are the rectangular coordinates of the point whose cylindrical coordinates are  

(2, Pi/4, 3)? [10 points] 

         x = __________ 

         y = __________ 

         z = __________

2.  What are the spherical coordinates of the point whose rectangular coordinates are  

(x = 1, y = 1, z = sqrt(2))? [10 points]

3.  For F = < [10 points] 

 b. Is F conservative?  Explain why or why not. 

c. Find div F = __________________ 

4. Evaluate the double integral over the region D, the triangular region with vertices (0, 4), (2, 0), and  

(0, 0). [10 points] 

       D 

5.  Using polar coordinates, evaluate the double integral over the region D, the disk .  

[10 points] 

      D

6. A lamina occupies the region bounded by   and and the density at each point is given by the function f(x,y) = 3xy.  What is the total mass? [10 points] 

7. Find the surface area of z = 2x + 3y that lies inside the cylinder [10 points]

8. Evaluate the triple integral over E where E is bounded by the parabolic cylinder and the planes y = 1, and z = 5 in the first octant.  [10 points] 

 E

9.  Sketch the following vector fields. [10 points] 

   a) F = <1, 1>                                                                                b) F = <x, y> 

10. Given F = is conservative:   [10 points] 

a) Find f such that the gradient of f = F.                            (Hint:  You can use the quicker method we have used in class)

b) Use the fundamental theorem of line integrals and your answer to part (a) to calculate the following integral where C is the curve going from (0, 0, 1) to (2, 4, 1)