MVC
I nstructions: SHOW ALL WORK !!
Quiz #7 NAME:
No calculator allowed.
1.
(6 pts) If the curve C is given by
   tan( ), ( )
 sec( ) with 0

4
, set up the integral

as an integral in terms of t .
C xy ds
2.
(6 pts) Convert the integral

D sin( x
2  y
2
) dA into an equivalent integral in polar coordinates if D is the region described by x y 4
 x
2
} .
MVC

Spherical/Cartesian



2  x
2  y
2  z
2



 x y z





 sin cos sin sin cos


 x
2  y
2


 tan
  tan
 z
 y x
3.
(6 pts) Convert the integral from rectangular coordinates into an equivalent integral in spherical coordinates, where D is the region in the first octant bounded by the planes y
 x and y

3 x , the cone z
 x
2  y
2
, and the spheres x
2  y
2  z
2 
2 and x
2  y
2  z
2 
8 .

D x x
2  y
2  z
2 dV
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
2  y
2
) i

2 xy j , find the circulation of F around 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
  as an integral or integrals in the single variable t
. DON’T
C
 integrate. b.
If ( , )


3 2 x y
 i
ˆ 
( x

)
ˆ
, determine a double integral that gives the flux across C of the vector field F
, making sure to provide appropriate limits of integration. DON’T integrate.
MVC

6.(8 pts) Determine whether Green’s theorem can be used to evaluate

C
Ln x
2  2
)

( x

6)
1
3 dy where C is the curve pictured. Explain carefully a. Circle of radius 3 centered at (3,3) b. Unit circle centered at origin
MVC c. C is the union of the two circles. d. Parallelogram with vertices (0,1), (2,1), (3, 3), and (1, 3) .

7.(8 pts) Using a suitable change of variables, evaluate

D
 x
2  y
2
 dx dy , where D is the region in the first quadrant of the xy -plane bounded by the graphs of y

,
 
1, y

1 x
, and y

2 x
.
MVC