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Chapter 17
 (x
2
 3xy  y )dx , where c is the arc
2
y  2x2 , 0  x  2 .
1.
Evaluate the line integral
2.
Consider the vector field F( x, y)  i  xj . If a particle starts at the point (10, 4) in the velocity field
given by F, find an equation of the path it follows.
3.
c
If a wire with linear density 10 lies along a space curve C, its moment of inertia about the z-axis is
given by I z 

10 ( x 2  y 2 ) ds . Find the moment of inertia for the wire in the shape of the helix
C
x  2 sin(t ), y  cos(t ), z  3t, 0  t  2
4.
Find a function f such that F  f , and use it to evaluate
 F  dr
along the given curve C.
C
F  e2 y i  (1  2xe2 y ) j ,
C : r(t )  tet i  (1  t ) j, 0  t  1
2r
where r = x i + y j + z k. Find the
| r |3
work done by F in moving an object from a point P1 along a path to a point P2 in terms of the
distances d1 and d 2 from these points to the origin.
5.
Suppose that F is an inverse square force field, that is, F 
6.
Use Green's Theorem to find the work done by the force F( x, y)  x( x  5 y)i  (4xy 2 ) j in moving a
particle from the origin along the x-axis to (3, 0) then along the line segment to (0, 3) and then back to
the origin along the y-axis.
Select the correct answer.
a.
7.
49.5
b.
-1.5
c.
4.5
d.
-4.5
e.
-49.5
Let r = xi + yj + zk and r = | r |.
Find   (9 r r) .
8.
Which of the equations below is an equation of a plane?
Select the correct answer.
a.
r (u, v)  (5  10u)i  (u  9v) j  (2  6u  4v)k
b.
r (u, v)  u cosvi  u sin vj  u k
2
9.
Find a parametric representation for the part of the sphere x 2  y 2  z 2  4 that lies above the cone
z  x2  y2 .
10. Find the area of the part of paraboloid x  y 2  z 2 that lies inside the cylinder z 2  y 2  25 .
11. Find the area of the part of the surface y  4x  z 2 that lies between the planes x = 0, x = 7, z = 0, and z = 3.
Select the correct answer.
a. 121.339
b. 113.269
12. Evaluate the surface integral.

c.
107.759
d.
110.234
e. 111.239
4( x 2 y  z 2 ) dS
S
S is the part of the cylinder x 2  y 2  9 between the planes z = 0 and z = 4.
13. Evaluate the surface integral  F  dS ,
where F( x, y, z)  x , and S is the part of the plane
S
x  y  z 1 in the first octant.
14. Find parametric equations for C, if C is the curve of intersection of the hyperbolic paraboloid
z  y 2  x 2 and the cylinder x 2  y 2  16 oriented counterclockwise as viewed from above.
15.
Find the moment of inertia about the z-axis of a thin funnel in the shape of a cone z  5 4 x 2  4 y 2 ,
1  z  4 , if its density function is  ( x, y, z)  3  z .
2
2
2
16. Use Gauss's Law to find the charge contained in the solid hemisphere x  y  z  81, z  0 , if the
electric field is E( x, y, z )  xi  yj  2 zk .
17. Use Stokes' Theorem to evaluate

curl F  dS .
S
F( x, y, z)  2xyzi  2xyj  2x 2 yzk
S consists of the top and the four sides (but not the bottom) of the cube with vertices (3, 3, 3)
oriented outward.
18. Use Stokes’ theorem to calculate the surface integral  F  dS .
S
F ( x, y , z )  (e x sin y )i  (e x cos y  z ) j  yk , and S is the part of the cone
1 z  2 .
19. Use Stokes' Theorem to evaluate

z 2  x 2  y 2 for which
F  dr .
C
F( x, y, z)  0.16xi  5 yj  6( y 2  x 2 )k
C is the boundary of the part of the paraboloid z  0.16  x 2  y 2 in the first octant. C is oriented
counterclockwise as viewed from above.
xz
20. Find the div F if F ( x, y, z )  e (cos yz i  sin yz j  k ) .
1.
2624
21
2.
y
3.
80 13
4.
e5  1
5.
 1
1
2
 
 d 2 d1 
6.
c
7.
36 r
8.
a
9.
x  x, z  4  x 2  y 2 , y  y, 2  y 2  x 2  4
10.
101 101  1
6
x2
 46
2
11. e
12.
512
13.
3
6
14.
x  4 cos(t ), y  4 sin(t ), z  16 cos(2t )
15.
0.0843
16.
1944  0
17. 0
18. 0
19. 0
20.
e xz (2 z cos yz  x)
1.
Find the gradient vector field of f ( x, y)  ln( x  2 y) .
2.
Find the gradient vector field of f ( x, y, z)  x cos
3.
Consider the vector field F(x, y) = i + xj. If a particle starts at the point (10, 4) in the velocity field
given by F, find an equation of the path it follows.
4.
Evaluate

2y
.
5z
xy 4 ds , where C is the right half of the circle x 2  y 2  25 .
C
5.
Evaluate

xy dx  ( x  y ) dy , where C consists of line segments from (0, 0) to (3, 0) and from (3,
C
0) to (2, 5).
6.
Evaluate

yz dy  xy dz , where C is given by x  10 t , y  3t , z  10t 2 , 0  t  1 .
C
Select the correct answer.
a.
193.93
225.25
b.
20.82
c.
208.23
d.
1,939.29
e.
7.
A thin wire is bent into the shape of a semicircle x 2  y 2  4, x  0 . If the linear density is 7, find the
exact mass of the wire.
8.
Find the curl of the vector field.
F( x, y, z)  5e x sin( y)i  3e x cos(y) j  8zk
9.
If a wire with linear density 6 lies along a space curve C, its moment of inertia about the z-axis is
given by I z 

6( x 2  y 2 ) ds . Find the moment of inertia for the wire in the shape of the helix
C
x  2sin(t ), y  2 cos(t ), z  3t, 0  t  2 .
Select the correct answer.
a.
e.
48 13
b.
48 13 2
c.
12 13
d.
24 11
12 11
10. Find the work done by the force field F(x, y) = 3xi + (3y + 10)j in moving an object along an arch of
the cycloid r (t )  (t  sin(t )) i  (1  cos(t )) j , 0  t  2 .
11. Find the work done by the force field F(x, y) =xsin(y)i + yj on a particle that moves along the parabola
y  x 2 from (-2, 4) to (1, 1).
12. Find the work done by the force field F( x, y, z )  xzi  yxj  zyk on a particle that moves along the
curve r (t )  t 2i  t 3 j  t 4k , 1  t  0 .
13. Determine whether or not F is a conservative vector field. If it is, find a function f such that F  f .
F  (4  2xy  ln x)i  x 2 j
14. Determine whether or not F is a conservative vector field. If it is, find a function f such that F  f .
F  (9 ye9 x  sin y)i  (e9 x  x cos y) j
15. Find a function f such that F  f and use it to evaluate

F  dr along the given curve C.
C
F  yi  ( x  2 y) j
C is the upper semicircle that starts at (1, 2) and ends at (5, 2).
16. Determine whether or not the vector field is conservative. If it is conservative, find a function f such
that F  f .
F( x, y, z )  10 xi  10 yj  10 zk
17. Find the work done by the force field F in moving an object from P to Q.
F( x, y)  x3 y 4i  x 4 y3 j ; P(0, 0), Q(3, 2)
18. Let F  f , where f ( x, y)  sin(x  8 y) .
Which of the following equations does the line segment from (0, 0) to (0,  ) satisfy?
Select the correct answer.
a.

F  dr  0
C
b.

F  dr  1
C
c.
none of these
9r
is an inverse square force field, where r = x i + y j + z k.
| r |3
Find the work done by F in moving an object from a point P1 along a path to a point P2 in terms of
the distances d1 and d 2 from these points to the origin.
19. Suppose that F 
2
20. Use Green's Theorem to find the work done by the force F( x, y)  x( x  4 y)i  4xy j in moving a
particle from the origin along the x-axis to (1, 0) then along the line segment to (0, 1) and then back to
the origin along the y-axis.
1.
1
2
i
j
x  2y
x  2y
2.
 2 y  2 x  2 y  2 xy  2 y 
cos i  sin  j  2 sin k
 5z  5z  5z  5z
 5z 
3.
y
4.
6250
5.
-5.83
6.
a
7.
1029
 cos(64)  sin(16)
5
8.
 2e x cos(y)k
9.
a
10.
6 2
11.
-8.1
12.
87
88
13.
4x  yx2  x(ln x  1)  K
14.
ye9 x  x sin( y)  K
15.
8
16.
x2
 46
2
f  5x 2  5 y 2  5z 2  K
17.
324
18.
a
19.
 1
1
9  
 d 2 d1 
20.
0.333333