Math 32B  Calculus of Several Variables
Final Exam  Hints for Practice Problems
I do recommend that you only come to this document as a last resort  the more time you spend thinking
about the problems on your own, the better you will learn and remember the material for the ﬁnal. Having
said that, though, I did make the practice problems somewhat challenging, and so hints to the problems are
hidden in this document. To view a particular hint, simply select and copy the space under the corresponding
problem, and paste it somewhere else (e.g. Notepad). You should then hopefully be able to view the hint,
without having all the other problems spoiled!
Problem 1
Sometimes integrals are easier when you change the order of the integrals.
Problem 2
Note that you can ﬁnd the mass of the solid by computing what mass it used to have, and subtracting
the mass that was cut out.
Problem 3
It is important to visualise the solid properly, so try to draw a decent sketch. It may help to know that
the solid has four corners, at (0
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 Winter '10
 Corbin
 Polar coordinate system, Stokes' theorem

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