Sample Exam Sample Exam Sample Exam
MATH 2210 Sample Exam #2
Directions: Show your work. Correct answers without relevant supporting work will not
receive any credit.
1. Know how to work problems #5 and #9 from section 12.1. (Be able to take the sample
points to be midpoints or any of the corners of the squares.)
2. (a) Draw the level curves for
xyyxf 4),(
together with the graph of
1624
22
yx
.
(b) Find the points on the graph of the ellipse
1624
22
yx
where
),( yxf
has a
maximum and where
),( yxf
has a minimum. Locate and label these points on your
graphs. (Hint: Use Lagrange Multipliers)
3. Evaluate each iterated integral.
(a)
5ln
0 0
y
yx
dxdyey
(b)
1
0 0
4
x
x
x
dxdydzyx
4. Sketch the region of integration, reverse the order of integration and evaluate.
8
0
2
3
y
x
dydxe
5. Sketch the region bounded by
)1(
3
xxy
and
3
yx
, then use a double integral to
find the area of the region.
6. Find the volume of the solid bounded by the paraboloids
22
33 yxz
and
22
100 yxz
using a triple integral.
7. Set up but DO NOT EVALUATE an integral for the area of the part of the surface
23
yxz
that lies over the region bounded by
xy
and
4
xy
. Simplify your
result.
8. Rewrite the following triple integral in two different ways by integrating with respect
to y first each time.
1
1
11
0
2
),,(
x
y
dxdydzzyxf
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