MVC
Sample Quiz
NAME:
Instructions: SHOW ALL WORK !!
We don’t really need a calculator, do we?
#1. Rewrite

4  x 2  y 2 dA using polar coordinates, where D is the region in the first quadrant
D
bounded by the graphs of y  0, y  x, and x 2  y 2  4 .
2 x
#2 Calculate
  ( x  y) dy dx , by making the change of variables
x  u  v and y  u  v.
0 0
#3 Look at the integral
  x
2
 y 2  dV where D is the region above the plane z  0 , inside the
D
sphere x  y  z  4 , and outside the cone z  x 2  y 2 .
2
2
2
a. Write this integral as an iterated integral (or integrals) in cylindrical coordinates.
b. Write this integral as an iterated integral (or integrals) in spherical coordinates.
1
#4. Convert the integral

1
y ln(4  x  y ) dy dx to an integral in u and v using the transformation
x
0
x  u  uv, y  uv .
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
#5.
integral

x cos( y ) ds as an integral in terms of t.
C
#6.
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
x2  y 2  4 .
MVC
#7.
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.
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.
#8. 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