Test 2: Multiple Integrals
Review for Test 2: Multiple Integrals
Multivariable Calculus
Format
• The exam will contain 6 problems (plus or minus 2) and will last 50 minutes.
• It is a paper and pencil exam.
• You will need to show your work.
• You may use a graphing calculator. However, you may not use a symbolic calculator such as the TI-89. If you do not bring an acceptable calculator, you may have to do without.
• You must be able to answer warm up questions and paraphrase mathematical quotes such as the quote by the Scottish mathematician George Crystal who wrote, “Every mathematical book that is worth reading must be read ‘backwards and forwards,’ if I may use the expression. I would modify Lagrange's advice a little and say, ‘Go on, but often return to strengthen your faith.’ When you come to a hard or dreary passage, pass it over; and come back to it after you have seen its importance or found the need for it further on.”
Basic Content.
• You are responsible for sections 15.1-4 and 15.6-9.
• In addition to the material covered in the class, you are responsible for all of the basic facts you have learned since kindergarten. These include the facts that Barack Obama was the President of the United States of America, [pic], and that 1/0 is undefined.
Course Objectives: The student will be able to … evaluate multiple integrals using multiple coordinate systems.
In Studying . . .
• You should be able to recreate every derivation done in class
• You need to know the vocabulary.
• You should be able to solve every example done in class.
• You should be able to solve every homework question (or at least set up the integrals).
Ideas that may help with test prep …
• The review assignment in WebAssign includes questions covering material from all of the material being tested. The problems are jumbled up.
• Review the most recent material first.
• Consider recopying your notes.
• Summarize your notes. Make note cards for important formulas and definitions. Set them aside once the definitions are known.
• Rework examples from the quiz, class, and homework questions (in this order).
• In general, review exercises are a good source of practice problems. However, we skipped enough sections in this chapter that this may not be the case for Chapter 14.
• Practice like you will play – do you know the material without your notes when the clock is running?
• Study with a friend to have more fun.
• Look to resources such as the MIT videos and Khan Academy to fill in holes.
• Show up at least five minutes early for the exam.
Below are a few comments/questions meant to highlight or clarify key points:
1. Non-symbolic graphing calculators are acceptable. Bring your own or borrow one from me.
2. I do not quiz on stuff like the difference between Type I and Type II regions – I just expect that you can do the problems.
3. What is the difference between a double and iterated integrals?
4. What are the named theorems (someone’s name attached)?
5. Some problems will likely ask you to set-up but not solve integrals. Make sure that you follow the directions.
6. Regarding drawings
a. This is not an art class – you are not expected to give me perfect pictures
b. I would expect that you could draw some basic three dimensional objects such as a prism in the first octant, a sphere, a paraboloid, a wedge of cheese, cone, or even intersections of these. Each of these is straightforward and you should be able to sketch a graph if need be.
c. You should be able to work from a given graph to set-up an integral.
7. Regarding applications:
a. You need to know how to find the mass and center of mass of an object (as in chapter 15), but you do not need to find the moments of inertia (second moments).
b. I will not ask you to work with a probability density function on the exam.
8. Regarding the Jacobian
a. A change of variables is generally helpful in two places:
i. The region over which you are integrating can be simplified thru a change in variables.
ii. The integrand can be simplified by a change in variables.
b. You need to understand the Jacobian’s relationship to the changes in coordinate systems of polar, cylindrical, and spherical coordinates (it is the scaling factor).
c. You need to know how to make substitutions in the integrand and differential element.
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