Sample Questions for Exam 3
Math 2374, Spring 2002 (modified somewhat for Fall2003)
rogness@math.umn.edu

This is just a bunch of other suggested problems for you to work on, mostly taken from your textbook. I haven't put any problems specifically for 5.5 here, but the material in there is used in many (most?) other sections covered on this exam. So you certainly should be comfortable with parametrized surfaces.

Questions for Section 5.6

4.  Evaluate [Graphics:Images/index_gr_9.gif], where M is the surface parametrized by [Graphics:Images/index_gr_10.gif], [Graphics:Images/index_gr_11.gif], [Graphics:Images/index_gr_12.gif].  (Hint: you should start by deciding if this is a surface integral of a vector field or a scalar function.

5.  Evaluate [Graphics:Images/index_gr_13.gif], where M is the portion of the hyperbolic paraboloid [Graphics:Images/index_gr_14.gif]bounded by the planes x=0, y=1, and y=x.

6.  Evaluate [Graphics:Images/index_gr_15.gif] where F=(x,y,z) and M is parametrized and oriented by [Graphics:Images/index_gr_16.gif].

7.  Evaluate [Graphics:Images/index_gr_17.gif] where [Graphics:Images/index_gr_18.gif] and M is parametrized and oriented by [Graphics:Images/index_gr_19.gif], where s goes from 0 to 2, and t goes from 0 to Pi..

Questions for Section 5.7

8.  Evaluate the following double integrals:

(a) [Graphics:Images/index_gr_20.gif], R is the region in the first quadrant bounded by the lines y=0, y=x, and the circles (centered at the origin) of radius [Graphics:Images/index_gr_21.gif]and 5.

(b)  [Graphics:Images/index_gr_22.gif], R is the annular region given by [Graphics:Images/index_gr_23.gif].

(c) [Graphics:Images/index_gr_24.gif].

(d)  [Graphics:Images/index_gr_25.gif] where R is the region bounded by [Graphics:Images/index_gr_26.gif]

Questions for Section 4.2

9.  Do as many of #1-#7 on page 229 as you need until you feel confident that you can compute the divergence and curl of any vector field on the exam.  Answers to the odd numbered problems are in the back of your book.  Bonus question: Interpret your answers.  (For some of these questions, the geometric interpretation will be hard, but it's certainly possible for #1 and #3, and probably #5 as well.)

Questions for Section 6.4

10.  Compute the following integrals using Stoke's Theorem:

(a)  [Graphics:Images/index_gr_27.gif]5yz dx [Graphics:Images/index_gr_28.gif] C is parametrized by f(t)=(cos t, sin t, cos t - sin t), [Graphics:Images/index_gr_29.gif].

(b)  [Graphics:Images/index_gr_30.gif], C is parametrized by f(t)=(sin 2t, cos t, sin t), [Graphics:Images/index_gr_31.gif].

Questions for Section 5.8

I'm running out of time to type these up -- sorry!

11. Cylindrical coordinates: do #1 and #5

12. Spherical coordinates: do #7, #9.

13. Other changes of coordinates: do #15, and review #13 and #14 from the homework.

Questions for Section 6.3

14.  Do number 3.

15.  Re-do #17 from section 5.6 using the divergence theorem.

16.  Do number 7.  Here's a translation from the weird notation: F=(2x,2xy,4xyz), and the bounds on the box are for x, y, and z, respectively.




Converted by Mathematica      April 29, 2002