MVC
Chapter 6 Quiz
Instructions: SHOW ALL WORK !!
Ya got your calculator, notes, and book.
NAME:
 2 2t 3 
xy
1. (4 pts) If x  t    t , t ,
 with 1  t  3 is a path and f ( x, y, z )  z  e , write the line

3 

integral

f ds as a single integral using the variable t. Do not evaluate.
x
 F ds as a single integral using the variable t where
and the path is given by x  t    2t  1, 2t  1, t 2  with 0  t  1.
2. (5 pts) Write the vector line integral
x
F ( x, y , z )  x i  y j  z k
2
Do not evaluate.
MVC
3. (8 pts) a. Determine which of the following line integrals is independent of path. Explain.
 ( x  3x
i.
2
y 2 ) dx  (2 x3 y  3y 2 ) dy in going from (0,0) to (1,2).
C
ii.
  sec (x)sin( y)  dx
2

 tan( x) cos( y)  dy
in going from (0, 0) to ( ,  )
C
 x2

iii.  ( xy  3x 2 z 2 )dx    y 3  z dy  (2 x 3 z  y )dz in going from (0, 0, 0) to (  1, 1, 1)
 2

C
.
b.
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Evaluate one of the above integrals by finding the scalar potential function and the using
the fundamental theorem of line integrals.
4. (6 pts) Find the flux of F( x, y)  ( x 2  y 2 ) i  2 xy j across the boundary of the unit square
with vertices (0, 0), (1, 0), (1,1), and (0,1) .
MVC
5. (8 pts) Let C be a curve in the plane starting at (0, 0) moving to (4, 0) along the x-axis, from
(4,0) to (3,1) along the parabola x  4  y 2 , and then returning to (0, 0) along the line x  3 y .
a. Parameterize the path to express the line integral
 y dx  x dy as an integral or integrals
C
in the single variable t. Do not evaluate.
b. Apply Green’s Theorem to the integral in part a. to obtain one or more double integrals,
making sure to provide appropriate limits of integration. Do not evaluate.
MVC
6. (8 pts) Determine whether Green’s theorem can be used to evaluate
 ln( x
2
 y 2 )dx  9   x 2  y 2  dy where C is the curve pictured. Explain carefully.
C
a.
b.
c. C is the union of the two circles.
d.
MVC
 6 x2 
 4 x3 
dx

C  5 y 2   5 y3  dy where C is the curve given by:
 x  2  3cos(t )

,0  t  .

2
 y  1  2sin(t )
7. (5 pts) Evaluate
MVC