MVC
Quiz #7
Instructions: SHOW ALL WORK !!
No calculator allowed.
NAME:
1. (6 pts) If the curve C is given by x  t   cos(t ), y(t )  sin(3t ) with   t   , set up the
integral

x 2 y ds as an integral in terms of t.
C
2. (6 pts) Convert the integral
  3x  2 y  dA into an equivalent integral in polar coordinates if
2
D
D is the region described by {( x, y) x  y  4  x2 } .
MVC
Spherical/Cartesian
 x   sin  cos 

 y   sin  sin 
 z   cos 


2
2
2
2
  x y z

x2  y 2

tan



z

y

tan  

x
3. (6 pts) Convert the integral from spherical coordinates into an equivalent integral in
rectangular coordinates.
 
2 4 sec(  )
 
0 0
4 sin()d  d  d 
0
4. (6 pts) Let C be the boundary of the unit square with vertices (0, 0), (1, 0), (1,1), and (0,1) .
Given the vector field F( x, y)  ( x 2  y 2 ) i  2 xy j , find the outward flux of F across the curve C.
MVC
5. (8 pts) Let C be a curve in the plane starting at (1,0), moving to (0,1) along the circle of radius 1
centered at the origin, and then returning to (1,0) along a straight line segment..
a. Parameterize the path (in two pieces, most likely) to express the line integral
3 2
 x y dx   x  y  dy as an integral or integrals in the single variable t. DON’T
C
integrate.
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.
MVC
pts) Determine whether Green’s theorem can be used to evaluate
1
2
2
C x2  y 2 dx  ( x  3)  y dy where C is the curve pictured. Explain carefully
6. (8
a. Circle of radius 3 centered at (3,3)
b. Unit circle centered at origin
c. C is the union of the two circles.
d. Parallelogram with vertices (0,1), (2,1), (3,3), and (1,3) .
MVC
x y
7.(8 pts) Evaluate
x y
 e dA , using a convenient change of variables if D is the region inside the
D
trapezoid with vertices (1, 0), (2, 0), (0, 2) and (0, 1) .
MVC