MVC
Chapter 6 Quiz
NAME:
Instructions: SHOW ALL WORK !!
We don’t really need a calculator, do we?
If the curve C is given by x  t   t , y (t )  t 2 with 0  t  2 and f(x, y) = xy , set up the
#1.
integral

f ds as an integral in terms of t.
C
#2.
Find a parameterization for each of the following curves:
a. The line segment from (1, 2, 4) to (  2, 2, 0)
b. The curve that starts at the point (2, 0, 0) and goes to the point ( 2, 0,3) along the
cylinder x 2  y 2  4 .
MVC
#3.
Let C be a curve in the plane starting at (0,0), moving to (1,1) along the curve y = x3, and
then returning to the origin along the straight line y = x.
a. Parameterize the path (in two pieces, most likely) to express the line integral
2
 2 x y dx  xy dy as an integral or integrals in the single variable t. DON’T integrate.
C
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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#3. (continued) Let C be a curve in the plane starting at (0,0), moving to (1,1) along the curve
y = x3, and then returning to the origin along the straight line y = x.
c. Given the vector field F( x, y)  2 x 2 y i  xy 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.
MVC
4. Determine whether Green’s theorem can be used to evaluate

y  x dx  y 2 dy where C is the
C
curve given (Explain your reasoning).
a. C : x  2cos t, y  2sin t;0  t  2 
b. C: x  cos t  1, y  sin t  1 ; 0  t  2
c. C: the square with vertices (1, 2), (4, 2), (4,5), and (1,5) traversed counter-clockwise.
MVC