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Name:
Math 380 – Exam 3
July 30, 2008
50 points possible
1.
(a) (3pts) State the conclusion to the Change of Variables Theorem for Double Integrals
and explain the role of the Jacobian factor in the integrand.
(b) (7pts) Determine an appropriate change of variables and evaluate the integral
ZZ
R
e
x

y
x
+
y
dA
where
R
is the rectangle bounded by the lines
y
=
x
,
y
=
x
+ 5,
y
= 2

x
and
y
= 4

x
.
(More space on following page if needed.)
1.
(b) Continued.
2.
Consider the change of variables
T
(
u,v,w
) =
(
u
2
+ 8
uw, v, u
2
w
+ 3
)
. Let
D
*
be the set
of (
u,v,w
)
∈
[

1
,
1]
×
[0
,
2]
×
[

1
,
0].
(a) (2pts) Find the Jacobian of
T
.
(b) (2pts) Are there any points in
D
*
where this change of variables may not be appropri
ate? Brieﬂy explain.
(c) (2pts) If
f
(
u,v,w
) is continuous on
D
*
and
f
(
T
(
u,v,w
)) is continuous on
D
=
T
(
D
*
),
explain why this change of variables is still valid and the integral will evaluate correctly.
3.
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 Summer '08
 Staff
 Math, Calculus, Integrals

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