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
ds 

 x(t ) 2   y(t ) 2 dt 
f ds 
C

C
#2.
1  (2t ) 2 dt 
2
2
   1  (2t ) dt   t
xyds  (t ) t
2
0
2
3
1  (2t ) 2 dt
0
Find a parameterization for each of the following curves:
a. The line segment from (1, 2, 4) to (  2, 2, 0)
 x  3t  1

 y  2 0  t 1
 z  4t  4

Many other possible correct answers
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 .

 x  2 cos t

 y  2sin t 0  t  

3t
 z


MVC
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.
#3.
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
 2x
2
y dx  xy dy 
C
 2x
2
y dx  xy dy 
 2x
2
y dx  xy dy
 C2
C1
1
1
  2t t dt  tt 3t dt   2t 2t dt  tt dt
2 3
3
0
2
0
1
  (2t 5  3t 6  2t 3  t 2 ) dt
0
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.
 2x
 N M 
y dx  xy dy    


x y 
0 x3 
1 x
2
C
 N M 
 


x y 
0 x3 
1 x
1 x
    y  2 x 2 dydx
0 x3
MVC
#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.
  N dx  M dy    xy dx  2 x
C
2
y dy
C

  xy dx  2 x
2
y dy 
  xy dx  2 x
2
y dy
 C2
C1
1
1
  tt dt  2t t 3t dt   tt dt  2t 2t dt
3
2 3
2
0
0
1
  (t 4  6t 7  t 2  2t 3 )dt
0
d. Apply the divergence form of Green’s theorem to obtain a double integral that would
calculate this flux. DON’T integrate.
1 x
  N dx  Mdy    div( F )dA
C
0 x3
1 x
 M N 


 dA
x y 
0 x3 
1 x
    4 xy  x dA
0 x3
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
a. C : x  2cos t, y  2sin t;0  t  2 
NO. M ( x, y)  y  x is not C1 throughout the circular disk. We can see that M is
undefined when x  y .
b. C: x  cos t  1, y  sin t  1 ; 0  t  2
YES. This circular path is oriented so as to keep the interior on the right. Both
M ( x, y)  y  x and N ( x, y)  y 2 are C1 throughout the circular disk and its
boundary, since the circle lies above the line y  x.
c. C: the square with vertices (1, 2), (4, 2), (4,5), and (1,5) traversed counter-clockwise.
NO. The line y  x runs through the region bounded by the square, so
M ( x, y)  y  x is not C1 throughout the square region.
MVC