Math 241 – Exam 4 – 2PM V1
Name:
December 8, 2008
Section Registered In:
55 points possible
1.
Consider a metal wire in the shape of the parabola
y
=
x
2
, 1
≤
x
≤
3.
(a)
(3pts) Find a parametrization of the wire.
(b)
(6pts) The density at a point (
x, y
) on the wire is given by
f
(
x, y
) = 3
y
x
g/cm. Assum
ing that the units on the
x
axis are in centimeters, compute the mass of the wire.
(c)
(2pts) If your parametrization in
C
was reoriented, how could that effect the integral
computed in part (b)? Briefly explain your answer.
2.
(a)
(5pts) Compute
Z
C
y dx

x dy
x
2
+
y
2
, where
C
is the circle of radius 1 centered at the origin,
oriented counterclockwise.
(b)
(3pts) State the formula at the conclusion of Green’s Theorem.
(c)
(3pts) Would Green’s Theorem have been an appropriate alternative to evaluating the
line integral in part (a)? Briefly explain your answer.
3.
(a)
(4pts) Let
S
be the sphere
x
2
+
y
2
+
z
2
= 16. Parametrize the sphere.
(b)
(7pts) Evaluate
ZZ
S
(
x
2
+
y
2
+
z
2
)
dS
.
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 Fall '08
 Kim
 Math, Vector Motors, Stokes' theorem, Surface normal, 3pts

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