Spring 2007, Math 236

April and May, 2007

Be sure to check back often, because assignments may change!

All section and page numbers refer to sections from Smith & Minton's Multivariable Calculus, 3rd edition.

Due Monday 4/2 at 9am

Section 13.1: Double Integrals

E-mail Subject Line: Math 236 Your Name 4/2

1. If f(x,y) is a positive function of two variables, what does _R f(x,y)dA measure?
2. Explain Fubini's Theorem in your own words. What is its importance?

Reminder:

• For Wednesday, remember that you either need to give me a progress report on your reading, or a hand-drawn sketch of your parametric picture, along with your Maple worksheet containing what you've got so far (at least 7 functions).

Due Wednesday 4/4 at 9am

Section 13.1: Double Integrals (continued)

• To read: Re-read the section. Pay careful attention to Example 1.7 -- I know it's at the end of a long section, but it is crucial.

E-mail Subject Line: Math 236 Your Name 4/4

In the double integral R f(x,y)dA,

1. What does dA represent?
2. Do the two integral signs go together -- that is, are they, in a sense, a new symbol? Or do they represent first one integral and then another?

Due Friday 4/6 at 9am

13.2: Area, Volume, and Center of Mass

E-mail Subject Line: Math 236 Your Name 4/6

1. If R is the circle of radius 3 centered at the point (5,1), what is R dA? (You should be able to do this without setting up any iterated integral)
2. Is My used to compute the x- or the y- coordinate of the center of mass? And in a related question -- why do you think it's called My?
Reminders:
• You don't have a problem set due next week because of the exams, but since of course I need to cover the rest of Section 13.1, Section 13.2, and and the first half of 13.3 (the sections that would normally appear on the next problem set) more thoroughly than I could with just a question or two on the exams, I have assigned some problems on these sections in the next problem set. If you wait until shortly before those problems are due, however, we will have built up on the material so much you may well find yourself completely lost, so I urge you (I beg you!) to do the problems from these sections as we finish them, especially Sections 13.1 and 13.2.

(If those arguments don't work for you, perhaps the length of next problem set will convince you! I cut the numbers down to what is probably an inadequate number, but the problem set will still be extremely long if you don't get started on it now.)

Due Monday 4/9 at 9am

Section 9.4: Polar Coordinates, Section 13.3: Double Integrals in Polar Coordinates

• To read: All of Section 9.4; through Example 3.2 in Section 13.3.

E-mail Subject Line: Math 236 Your Name 4/9

1. Find one polar coordinate representation for the rectangular point (5,-2).
2. Find rectangular coordinates for the polar point (1,Pi/3).
3. The graphs of (a) r=sin(7*theta) and of (b) r=sin(7*theta)+5 are shown below. Match the functions to the graphs.
Note: To get these, I used the commands
• polarplot(sin(7*theta),theta=0..Pi);
• polarplot(sin(7*theta)+5, theta=0..2*Pi);

Due Wednesday 4/11 at 9am

Exam 2 - In Class Portion

Due Friday 4/13 at 9am

Section 13.3: Double Integrals in Polar Coordinates

E-mail Subject Line: Math 236 Your Name 4/13

1. Why might we want to convert a double integral from rectangular coordinates to polar coordinates?
2. When we use iterated integrals to evaluate an integral in polar coordinates, what is dA?

Due Monday 4/16 at 9am

Section 13.4: Surface Area

E-mail Subject Line: Math 236 Your Name 4/16

1. Give a real-world example where you would want to compute the surface area.
2. After partitioning the region R, what objects are used to approximate the surface area over each subregion Ri?
3. Where does the cross-product come in?
4. What formula from earlier in the semester does the formula for surface area remind you of? Why might these two quantities have similar formulae?

Reminders:

• Remember to continue working on your individual project.

Due Wednesday 4/18 at 9am

Section 14.1: Vector Fields

E-mail Subject Line: Math 236 Your Name 4/18

1. Figures 14.3(a) and 14.3(b), shown below, are both representations of the vector field H(x,y)=. Which one is (probably) more accurate?
2. Suppose the vector field shown below represents the flow of a small portion of a river, and suppose a leaf were to fall into the water at the point (-2,2). Describe the movement of the leaf.

fieldplot(f(x,y), x=-2..3, y=-2..3, fieldstrength=maximal(2),color=red);

3. Compare this behavior with the path the leaf would follow if it fell in at the point (-2,2.5) instead.

Due Friday 4/20 at 9am

Section 14.2: Line Integrals

• To read: Through Theorem 2.3, top of page 1136.

E-mail Subject Line: Math 236 Your Name 4/20

1. Give two different uses for line integrals with respect to arc length.
2. If you're evaluating a line integral with respect to arclength, does the orientation of the curve matter? If so, what difference does orientation make?

Due Monday 4/23 at 9am

Section 14.2: Line Integrals (continued)

E-mail Subject Line: Math 236 Your Name 4/23

1. Give at least one use for component-wise line integrals.
2. If you're evaluating a line integral with respect to any of the axes, does orientation matter? If so, what difference does orientation make?

Reminders:

• Your individual project is due Friday -- you should be entering the final stages!

Due Wednesday 4/25 at 9am

Section 14.3: Independence of Path and Conservative Vector Fields

E-mail Subject Line: Math 236 Your Name 4/25

1. Why are conservative vector fields so helpful when evaluating line integrals?
2. Why is Theorem 3.2 called the Fundamental Theorem for line integrals?

Due Friday 4/27 at 9am

Section 14.4: Green's Theorem

E-mail Subject Line: Math 236 Your Name 4/27

1. What surprises you about Green's Theorem?
2. Give an example of a region R in the plane where Green's Theorem does not hold.

Reminders:

• Project due Friday!

Due Monday 4/30 at 9am

Section 14.4: Green's Theorem (continued)

Due Wednesday 5/2 at 9am

Section 13.5: Triple Integrals

E-mail Subject Line: Math 236 Your Name 5/2

1. Why might you need to compute a triple integral?
2. What is often the hardest part of calculating a triple integral?

Due Friday 5/4 at 9am

Section 13.6: Cylindrical Coordinates

E-mail Subject Line: Math 236 Your Name 5/2

1. What are some of the advantages to using cylindrical coordinates?

Here ends the reading for the semester!

Janice Sklensky
Wheaton College
Department of Mathematics and Computer Science
Science Center, Room 109
Norton, Massachusetts 02766-0930
TEL (508) 286-3973
FAX (508) 285-8278
jsklensk@wheatonma.edu

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