Test 3 sample exam
If you have any questions, or think you’ve spotted an error / typo, please get in
touch with me. Some of the problems are a little tricky, but its good practice
(and better I get ‘rid of’ those problems before the actual test).
1.
(a) Find and classify the critical points of
f
(
x, y
) =
x
3

3
x
+
y
2
.
(6
points)
(b) Use your answer to part (a) to find the global minimum and max
imum of this
f
on the closed disk, radius 1 and center (1
,
0).
(8
points)
2. Find the volume of the region above the square with vertices (1
,
1), (

1
,
1),
(1
,

1), (

1
,

1) and below the graph of the function
f
(
x, y
) = 3
x
2
+6
y
2
.
(5 points)
3. Evaluate
Z
1
0
Z
1
√
y
3 cos(
x
3
)
dxdy.
(8 points)
4.
(a) Show that
Z
∞
∞
Z
∞
∞
e

x
2

y
2
dxdy
=
Z
∞
∞
e

x
2
dx
2
.
(3 points)
(b) Use your answer to part (a) (and a shift to polar coordinates on the
left hand side) to show that
Z
∞
∞
e

x
2
dx
=
√
π.
(7 points)
5. Let
R
be the region in the positive octant, inside the cylinder
x
2
+
y
2
= 9
and below the paraboloid
z
=
x
2
+
y
2
. Using whichever coordinate system
will give the simplest result, write down
(a) a double integral (4 points), and
(b) a triple integral (4 points)
giving the volume of
R
. Find the volume of
R
. (3 points)
6. Let
f
(
x, y, z
) be a function of three variables. Write down the integral of
f
over the region in the positive octant bounded by the parabolic cylinder
x
= 1

y
2
and the plane
z
=
y
+ 2 in the orders
(a)
dzdydx
(4 points), and
1
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(b)
dxdzdy
(4 points).
7. Let
R
be the region consisting of all points in the first octant which are
at distance at most 2 from the origin. Write down triple integrals giving
the volume of
R
in:
(a) Cartesian coordinates (4 points);
(b) cylindrical coordinates (4 points);
(c) spherical coordinates (4 points).
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 Spring '08
 ALDROUBI
 Coordinate system, Spherical coordinate system, Multiple integral, Polar coordinate system

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