MVC
Chapter 6,7 Sample Quiz
Instructions: SHOW ALL WORK !!
Calculator Allowed.


1. If x  t   cos t ,sin t , t 2 with 0  t   is a path and f ( x, y, z )  x  e yz , write the line
integral

f ds as a single integral using the variable t.
x
2. Evaluate the vector line integral


 F ds where F ( x, y, z)  xi  yj  zk and the path is
x
given by x  t   2t  1, t ,2t  1 with 0  t  1.
2
3. Parameterize the portion of the surface r   , 0    6 , in cylindrical coordinates between
the planes z  2 and z  2 . Give the parameterization in terms of x,y, and z.
4. a.
i.
Determine which of the following line integrals is independent of path in going from
(0,0) to (1,2).
 ( x  3x
2
y 2 ) dx  (2 x3 y  3y 2 ) dy
C
ii.  sin x cos y dx  sin x sin y dy
C
iii.
(2 xy 5  2xy 2 )
(5 y 4  2y )
dx

dy
 ( x 2  1)2
x2  1
C
b.
Evaluate one of the above integrals by finding the scalar potential function and the using
the fundamental theorem of line integrals.
5. Let C be a curve in the plane starting at (1,0) moving to (1,0) along the curve 1  x 2 , and
then returning to (1,0) along the x-axis.
a. Parameterize the path to express the line integral
 ( x  y )dx  x
2
y dy as an integral or
C
integrals in the single variable t. DON’T integrate.
b. Apply Green’s Theorem to the integral in part a. to obtain a double integral, making
sure to provide appropriate limits of integration. DON’T integrate.
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5. (continued) Let C be a curve in the plane starting at (1,0) moving to (1,0) along the curve
1  x 2 , and then returning to (1,0) along the x-axis.
c. Given the vector field F( x, y)  ( x  y) i  x 2 y j , write the integral(s) in the single
variable t you would need to evaluate to find the outward flux of F across the curve C.
DON’T integrate.
d. Apply the divergence form of Green’s theorem to obtain a double integral that would
calculate this flux. DON’T integrate.
6.
Determine
whether
Green’s
theorem
can
be
used
to
evaluate
1
2
2
C ( x  3)2  ( y  3)2 dx  ( x  3)  ( y  3) dy where C is the curve pictured. Explain
carefully:
a. Circle of radius 3 centered at
b. Unit circle centered at origin
(3,3)

c. C is the union of the two circles.

d. Parallelogram
with
(1,0),(1,0),(3,3), and (1,3)
vertices
7. Set up (but do not evaluate) a double integral that gives the surface area of the surface
parameterized by X( s, t )   t ( s 2  t 2 ), s, s 2  t 2  , 0  s  1,0  t  1.
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8. Let S be the portion of the surface x 2  y 2  z 2  2 above the plane z  1 . Orient S upward and
give it’s boundary, C, a consistent orientation.
a. Parametrize C and use the parametrization to evaluate  F  ds , where F is the
C
vector fieled F  ( xz, y, y ) .
b. Use Stokes’s Theorem to write down an equivalent double (iterated) integral.
c. Find an integral that can be used to compute the upward flux across S.
9. Calculate the outward flux of F  i  j  k across the closed surface x 4  y 4  z 4  1 .
10. Let S be the part of the surface z  xy with x 2  y 2  1 . Calculate the upward flux of
F  ( y, x, z ) across the surface S .
11. Find
 yz dS
where S is the part of the plane z  y  3 that lies inside the cylinder x 2  y 2  1.
S
 F dS , where
12. Evaluate
S
1


F ( x, y, z )   z 2 x, y 3  tan( z ), x 2 z  y 2  and S is the top half of the
3


sphere x  y 2  z 2  1 oriented with upward normal.
2
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