Practice exam for calc IV, Summer 2007 Exam 1
Exam 1 will consist of roughly 100 points of similar problems to those below.
No notes,
texts or calculators will be allowed on the exam. You will be provided with a formula sheet.
Answers to these problems will appear in a few days.
1. (15 points) a) Compute
1

3
2

x
2
2
x

1
x
2
dydx
.
b) Sketch the region in
R
2
over which the above integral is computed.
c) Write the above integral as a sum of one or more integrals in
dxdy
order. You are not
asked to compute the result!
2.
(15 points) In this problem you will compute
R
(
x

y
2
)
100
dA
where
R
is the region
in
R
2
bounded by
y
= 0
, y
= 1
, y
=
√
x,
and
y
=
√
x

1.
a) Sketch the region in
R
in the plane.
b) Guess a transformation
T
from (
u, v
) to (
x, y
) which will greatly simplify the integral.
Hint: finding the inverse transformation
T

1
is often easier
c) Sketch the region in (
u, v
) space which corresponds to
R
in (
x, y
) space.
d) Compute the Jacobian of
T
.
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 Spring '11
 Peters
 Calculus, Region, right circular cylinder, uniform mass density

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