Stewart – Calculus ET 8e Chapter 16 Form A
1. Find the work done by the force field in moving an object along an arch
of the cycloid
2. Find the exact value of where C is the line segment from (0, 0, 0) to (1, , ).
3. Determine whether or not vector field is conservative. If it is conservative, find a function f such
that
4. Find the curl of the vector field.
5. Find the curl of the vector field.
6. Determine whether or not vector field is conservative. If it is conservative, find a function f such
that
7. Use Stoke’s theorem to evaluate
C is the curve of intersection of the plane z = x + 9 and the cylinder
8. Use Stoke’s theorem to evaluate
C is the curve of intersection of the hyperbolic paraboloid and the cylinder
oriented counterclockwise as viewed from above.
9. Find parametric equations for C, if C is the curve of intersection of the hyperbolic paraboloid
and the cylinder oriented counterclockwise as viewed from above.
10. Find the area of the part of paraboloid that lies inside the cylinder
Stewart – Calculus ET 8e Chapter 16 Form A
11. Evaluate the line integral over the given curve C.
, where C is the line segment joining (–4, –5) to (5, 4)
12. Evaluate the line integral over the given curve C.
, where C is the line segment joining (–2, –1) to (4, 5)
13. A thin wire in the shape of a quarter-circle , , has a linear mass density
. Find the mass and the location of the center of mass of the wire.
14. The plot of a vector field is shown below. A particle is moved . By
inspection, determine whether the work done by F on the particle is positive, negative, or zero.
15. Determine whether F is conservative. If so, find a function f such that .
16. Determine whether F is conservative. If so, find a function f such that .
17. Let f be a scalar field. Determine whether the expression is meaningful. If so, state whether the
expression represents a scalar field or a vector field.
Stewart – Calculus ET 8e Chapter 16 Form A
18. Find the area of the surface S where S is the part of the surface that lies inside the cylinder
19. Find the area of the surface S where S is the part of the sphere that lies to the right of
the xz-plane and inside the cylinder
20. Find a vector representation for the surface.
The plane that passes through the point and contains the vectors and
.
Stewart – Calculus ET 8e Chapter 16 Form A
Answer Key
Stewart – Calculus ET 8e Chapter 15 Form B
1. Find the gradient vector field of
2. Consider the vector field
If a particle starts at the point in the velocity field given by F, find an equation of the path
it follows.
3. Find the work done by the force field in moving an object along an
arch of the cycloid
4. Calculate the work done by the force field
when a particle moves under its influence around the edge of the part of the sphere
that lies in the first octant, in a counterclockwise direction as viewed from above.
5. Determine whether or not F is a conservative vector field. If it is, find a function f such that
6. Find a function f such that , and use it to evaluate along the given curve C.
7. Find a function f such that and use it to evaluate along the given curve C.
C is the upper semicircle that starts at (1, 2) and ends at (5, 2).
Stewart – Calculus ET 8e Chapter 15 Form B
8. Use Green’s Theorem to find the work done by the force
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.
9. Evaluate the line integral.
10. Find the curl of .
11. Determine whether or not vector field is conservative. If it is conservative, find a function f such
that
12. Determine whether or not vector field is conservative. If it is conservative, find a function f such
that
13. Find the moment of inertia about the z-axis of a thin funnel in the shape of a cone
if its density function is
14. Use Stoke’s theorem to evaluate
C is the curve of intersection of the hyperbolic paraboloid and the cylinder
oriented counterclockwise as viewed from above.
15. Find a parametric representation for the part of the plane that lies inside the cylinder
16. Evaluate the line integral over the given curve C.
, where C is the line segment joining (–5, –7) to (3, 1)
Stewart – Calculus ET 8e Chapter 15 Form B
17. Determine whether F is conservative. If so, find a function f such that .
18. Find (a) the divergence and (b) the curl of the vector field F.
19. Let f be a scalar field. Determine whether the expression is meaningful. If so, state whether the
expression represents a scalar field or a vector field.
20. Use Stokes’ Theorem to evaluate .
;
S is the part of the ellipsoid lying above the xy-plane and oriented with
normal pointing upward.
Stewart – Calculus ET 8e Chapter 15 Form B
Answer Key
Stewart – Calculus ET 8e Chapter 15 Form C
Select the correct answer for each question.
____ 1. Find the gradient vector field of f.
a.
b.
c.
d.
e. None of these
____ 2. Find the gradient vector field of the scalar function f. (That is, find the conservative vector field F
for the potential function f of F.)
a.
b.
c.
d.
Stewart – Calculus ET 8e Chapter 15 Form C
____ 3. Evaluate the line integral
where and C is the arc of the circle traversed
counterclockwise from ( , 0) to (0, ). Round your answer to two decimal places.
a.
b.
c.
d.
e.
____ 4. Find the exact mass of a thin wire in the shape of the helix
if the density is 5.
a.
b.
c.
d.
e.
____ 5. Find the work done by the force field on a particle that moves along the parabola
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 15 Form C
____ 6. Evaluate the line integral over the given curve C.
; ,
a.
9
2
b.
69
2
c.
231
2
d.
81
2
____ 7. Evaluate the line integral over the given curve C.
; ,
a.
28
b.
26
c.
40
d.
97
3
____ 8. Evaluate , where C is given by Round your
answer to two decimal place.
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 15 Form C
____ 9. Show that F is conservative, and find a function f such that , and use the result to evaluate
, where C is any curve from to .
; and
a. 0
b. –154
c. –107
d. –89
____ 10. Let D be a region bounded by a simple closed path C in the xy. Then the coordinates of the
centroid where A is the area of D.
Find the centroid of the triangle with vertices (0, 0), ( , 0) and (0, ).
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 15 Form C
____ 11. Use Green’s Theorem to evaluate the line integral along the positively oriented closed curve C.
, where C is the cardioid .
a.
20
b. –80
c.
10
d.
40
____ 12. A plane lamina with constant density occupies a region in the xy-plane bounded by a
simple closed path C. Its moments of inertia about the axes are
Find the moments of inertia about the axes, if C is a rectangle with vertices (0, 0), (4, 0),
(4, 5) and .
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 15 Form C
____ 13. Below is given the plot of a vector field F in the xy-plane. (The z-component of F is 0.) By
studying the plot, determine whether div F is positive, negative, or zero.
a. cannot be determined
b. negative
c. zero
d. positive
____ 14. Find the curl of the vector field F.
a.
b.
c.
d.
____ 15. Suppose that where g is a function of one variable such that
.
Evaluate where S is the sphere
a.
b.
c.
d.
e. None of these
Stewart – Calculus ET 8e Chapter 15 Form C
____ 16. Evaluate .
; S is the part of the plane in the first octant.
a.
72
29
b. 0
c.
403
d.
273
____ 17. Evaluate .
; S is the part of the cone between the planes and
.
a.
b.
c.
70
3
d. 0
____ 18. Evaluate .
; S is the part of the torus with vector representation
, , .
a. 0
b.
c.
d.
Stewart – Calculus ET 8e Chapter 15 Form C
____ 19. Find the mass of the surface S having the given mass density.
S is part of the plane in the first octant; the density at a point P on S is equal to the
square of the distance between P and the xy-plane.
a. 42
b.
c. 19
d.
____ 20. Let S be the cube with vertices . Approximate by using a
Riemann sum as in Definition 1, taking the patches to be the squares that are the faces of the
cube and the points to be the centers of the squares.
a.
b.
c.
d.
e. none of these
Stewart – Calculus ET 8e Chapter 15 Form C
Answer Key
Stewart – Calculus ET 8e Chapter 16 Form D
Select the correct answer for each question.
____ 1. Which plot illustrates the vector field
a.
c.
b.
d.
____ 2. Find the exact mass of a thin wire in the shape of the helix
if the density is 5.
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 16 Form D
____ 3. Find the work done by the force field F on a particle that moves along the curve C.
; ,
a.
17
b.
19
c.
34
d.
71
2
____ 4. Evaluate where C is the right half of the circle
a.
b.
c.
d.
e.
____ 5. Evaluate for the vector field F and the path C. (Hint: Show that F is conservative, and
pick a simpler path.)
C:
a. 24
b. 47
c. 0
d.
Stewart – Calculus ET 8e Chapter 16 Form D
____ 6. Show that F is conservative, and find a function f such that , and use the result to evaluate
, where C is any curve from to .
; and
a. –89
b. 0
c. –154
d. –107
____ 7. Show that F is conservative and find a function f such that , and use this result to evaluate
, where C is any path from to .
; and
a.
b.
c.
d.
____ 8. Use Green’s Theorem and/or a computer algebra system to evaluate
where C is the circle with counterclockwise orientation.
a.
b.
c.
d.
e. None of these