Exam #3 Review

Scheduling Information:

 Time: Thursday, April 13, 7:30-8:20AM.
Location: DH 2315 (9:30 lecture) and DH 2210 (12:30 lecture).
Review Session: TBA.
 

Topics:

 Double Integrals over Rectangles (15.1)
Double Integrals over General Regions (15.2)
Double Integrals in Polar Coordinates (15.3)
Surface Area (15.5)
Triple Integrals (15.6)
Triple Integrals In Cylindrical Coordinates (15.7)
Triple Integrals in Spherical Coordinates (15.8)
Change of Variables in Multiple Integrals (15.9)

 

Review Quesionts:
  1. What is the double integral of a function over a rectangle? How is it defined? How is it computed?
  2. What is an iterated integral? How is one related to a double integral?
  3. What does Fubini's theorem say?
  4. How can you find the average value of a function of two variables over a rectangle R?
  5. How do we define the double integral of a function over general region? How is it related to double integrals over rectangles?
  6. What is a region of type I? Of type II? Why do we care about these things?
  7. What "properties of double integrals" are listed in your textbook? Which ones have you used this semester?
  8. How can we integrate over a region that is neither of type I nor of type II?
  9. What is a polar rectangle? How can you write a double integral as an iterated integral with respect to polar coordinates? What must you never forget?
  10. Why might you prefer to compute a double integral using polar coordinates? (There two good answers) [easier to find limits, easier to compute antiderivatives]
  11. How can you use a double integral to find the area of a surface?
  12. What is a triple integral? How is it defined? How is it computed?
  13. What does Fubini's theorem tell you about triple integrals?
  14. What is a region of type 1? Of type 2? Of type 3?
  15. What are cylindrical coordinates?
  16. In what situations is it often adventageous to use cylindrical coordinates?
  17. How do you compute a triple integral as an iterated integral using cylindrical coordinates?
  18. What are spherical coordinates?
  19. In what situations is it often adventageous to use spherical coordinates?
  20. How do you compute a triple integral as an iterated integral using spherical coordinates?
  21. What is the Jacobian of a transformation x=g(x,y), y=h(x,y)? Of the transformation x=g(u,v,w), y=h(u,v,w), z=k(u,v,w)?
  22. What do Jacobians have to do with multiple integrals?
  23. What do Jacobians have to do with the expressions we've seen for integrals using polar, cylindrical, or spherical coordinates?
 

Exercises:

 Section 15.1 #1, 5, 9, 11, 13, 15, 21, 25, 29, 31, 35, 39, 47.
Section 15.2 #5, 9, 11, 15, 17, 21, 25, 47, 57.
Section 15.3 #3, 5, 11, 13, 15, 25, 31, 39.
Section 15.5 #3, 9, 11, 21, 23.
Section 15.6 #3, 7, 9, 13, 19, 21, 23, 27, 33, 35, 53.
Section 15.7 #7, 9, 15, 17, 21, 23, 29.
Section 15.8 #1, 3, 5, 7, 9, 13, 15, 17, 19, 21, 25, 27, 41.
Section 15.9 #5, 7, 11, 17, 23, 25, 27.