Mock Exam 4 – Version 1. December 1, 2009 Tuesday
This is longer than the actual 50minute exam, but work on as many as possible on your
own. The actual exam will have 6 to 8 problems,
all
of which you should complete.
1.
(a)
State the conclusion to the Change of Variables Theorem for Double Integrals and
explain the role of the Jacobian. Why is there an absolute value around the Jacobian in the
Change of Variables Theorem for Double Integrals?
(b)
Determine an appropriate change of variables and evaluate the integral
ZZ
D
r
x
+
y
x

2
y
dA
where
D
is the region in
R
2
enclosed by the lines
y
=
x/
2,
y
= 0, and
x
+
y
= 1.
2.
Express the integral
Z
3
0
Z
x
0
dydx
p
x
2
+
y
2
as an iterated integral in polar coordinates.
3.
Let
W
be the region in
R
3
above the
xy
plane bounded by the two spheres
x
2
+
y
2
+
z
2
= 9
and
x
2
+
y
2
+
z
2
= 25. Set up and then evaluate an iterated integral(s) that will calculate
the volume of
W
.
4.
[This is a review problem in order to help you distinguish scalar line integrals, vector line
integrals, scalar surface integrals, and vector surface integrals.] Consider a metal wire in the
first quadrant in the shape of the circle
x
2
+
y
2
= 4.
(a)
Find a parametrization of the wire.
(b)
The density at a point (
x, y
) on the wire is given by
f
(
x, y
) = 4
x

3
y
g/cm. Compute
the mass of the wire.
(c)
If your parametrization in
C
was reoriented, how could that effect the integral computed
in part (b)? Briefly explain your answer.
(d)
The integral in part (b) is called
ca
ar line integral. It computes m
s, where
f
(
x, y
) =
4
x

3
y
is called a d
n
ty function.
5.
(a)
Compute
Z
C
ydx

xdy
x
2
+
y
2
, where
C
is the circle of radius 1 centered at the origin, oriented
counterclockwise.
(b)
State the formula at the conclusion of Green’s Theorem.
(c)
Would Green’s Theorem have been an appropriate alternative to evaluating the line
integral in part (a)? Briefly explain your answer.
(d)
The integral in part (a) is called
e
or line integral.
It computes w
k.
We say
F
(
x, y
) = (
,
) is a f
ce field function.
6.
(a)
Let
S
be the sphere
x
2
+
y
2
+
z
2
= 16. Parametrize the sphere.
(b)
Evaluate
ZZ
S
(
x
2
+
y
2
+
z
2
)
dS
.
(c)
The integral in part (b) is called
ca
ar surface integral.
It computes m
s, where
f
(
x, y, z
) = (
x
2
+
y
2
+
z
2
) is called a d
n
ty function.
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7.
Let
S
be the part of the cylinder
x
2
+
z
2
= 4 bounded by

1
/
2
≤
y
≤
1
/
2.
(a)
Find a parametrization of
S
.
(b)
Find the general formula for the normal vector.
(c)
Let
F
(
x, y, z
) = (
x,
sin
y, z
). Compute the flux of the vector field
F
across the surface
S
.
(d)
The integral in part (c) is called
e
or surface integral. It computes f
x. We say
F
is
a v
tor field function.
(e)
If
T
is the part of the cylinder
x
2
+
z
2
= 4 bounded by the sphere
x
2
+
y
2
+
z
2
= 1
/
4,
what is the parametrization of
T
?
8.
(a)
State the formula at the conclusion of Stokes’ Theorem.
(b)
Evaluate
RR
S
(
∇ ×
F
)
·
ndS
, where
F
(
x, y, z
) = (
e
z
2
,
4
x

y,
8
z
sin
y
) and where
S
is
the portion of the paraboloid
z
= 4

x
2

y
2
above the
xy
plane, oriented so that the unit
normal vectors point to the outside of the paraboloid.