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Chapter 6 Quiz
NAME:
Instructions: SHOW ALL WORK !!
We don’t really need a calculator, do we?

2t 3 
Let the curve C be given by:  x(t ), y(t ), z (t )    t , t 2 ,
 with 1  t  2 . Set up each
3 

integral below as a single integral in terms of t.
#1.
a.

xyz ds
b.

F  d s given Fx, y, z = zi + yj + xk.
C
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 (0, 0, 0) and goes to the point (1, 0,1) along the circular
paraboloid z  x 2  y 2 .
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#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.
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#4 Use Green’s Theorem (with a thoughtful choice of F ( x, y )  M ( x, y )iˆ  N ( x, y ) ˆj ) to set up a line
integral that gives the area of the region inside the curve C : x(t )  t sin t , y (t )  3cos t;0  t  2 .
Write the line integral in terms of t. [Note that C is simple, smooth, closed curve].
#5. Set up a double integral that represents the area of the parametrized surface:
X  (2s  3t  1, t , s  t ); 1  s  1,0  t  1
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x2  y 2
#6. Find a parametrization for the portion of the surface z 
that lies above the region in the
x2
1
1
4
xy-plane bounded by the curves y  x, y  3x, y  , and y  . .
3
x
x
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