Math 32B  Calculus of Several Variables
Final Exam  Practice Problems
Disclaimer :
The choice of problems was entirely my own. I have no knowledge of what will appear on the
midterm, nor how the exam will be structured. These questions are designed to cover what I think are some
of the important parts of the course so far, but I do not claim that they cover everything you may be asked
in the midterm. Finally, any mistakes are my own entirely.
Problem 1
Evaluate the double integral
20
0
10
1
2
y
e

x
2
dx dy
.
Problem 2
A solid sphere of radius 10 has an ellipsoid cut out of its middle.
The ellipsoid has radii (distance from
the origin) of 2, 3, and 6 in the
x
,
y
and
z
directions respectively. If the density at the point (
x, y, z
) is
z
2
,
find the mass of the resulting solid.
Problem 3
A homogeneous solid of density 1 is bounded by the surfaces
x
= 1,
y
=
x
2
,
y
+
z
= 1 and
z

y
= 1.
Represent the mass of the solid as a triple integral, and write down the iterated integrals for all six possible
orders of integration. Then choose one and compute the mass.
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 Winter '10
 Corbin
 Cos, e−x dx dy

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