Stewart – Calculus ET 8e Chapter 16 Form D
____ 9. Use Green’s Theorem to evaluate the line integral along the positively oriented closed curve C.
, where C is the triangle with vertices , , and .
a.
6
b.
60
c.
30
d.
66
____ 10. Use Green’s Theorem to find the work done by the force
in moving a particle in the positive direction once around the triangle with vertices , ,
and .
a. 7
b.
7
6
c.
7
3
d. 42
____ 11. 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 16 Form D
____ 12. Find the correct identity, if f is a scalar field, F and G are vector fields.
a.
b.
c.
d. None of these
____ 13. Let
a. 5
b. 3
c. 1
d. 2
e. None of these
____ 14. Evaluate the surface integral. Round your answer to four decimal places.
S is surface
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 16 Form D
____ 15. 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
____ 16. Use Stokes’ Theorem to evaluate .
;
C is the curve obtained by intersecting the cylinder with the hyperbolic paraboloid
, oriented in a counterclockwise direction when viewed from above
a.
b. 0
c.
d.
1
Stewart – Calculus ET 8e Chapter 16 Form D
____ 17. Use a computer algebra system to compute the flux of F across S. S is the surface of the cube cut
from the first octant by the planes
a.
b.
c.
d.
e.
____ 18. Assuming that S satisfies the conditions of the Divergence Theorem and the scalar functions and
components of the vector fields have continuous second order partial derivatives, find
,
where a is the constant vector.
a. 8
b. 6
c. 5
d. 7
e. 0
Stewart – Calculus ET 8e Chapter 16 Form D
____ 19. Find the area of the surface.
The part of the paraboloid ; ,
a.
b.
c.
d.
____ 20. Find a parametric representation for the part of the elliptic paraboloid that lies in
front of the plane x = 0.
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 16 Form D
Answer Key
Stewart – Calculus ET 8e Chapter 16 Form E
____ 1. Match the vector field with its plot. Select the correct answer.
a.
c.
b.
d.
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.)
Stewart – Calculus ET 8e Chapter 16 Form E
____ 3. Find the exact mass of a thin wire in the shape of the helix
if the density is 5.
Select the correct answer.
a.
b.
c.
d.
e.
4. Evaluate the line integral over the given curve C.
; ,
____ 5. A thin wire is bent into the shape of a semicircle If the linear density is , find
the exact mass of the wire.
Select the correct answer.
a.
b.
c.
d.
e.
____ 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
Select the correct answer.
a. –89
b. 0
c. –154
d. –107
Stewart – Calculus ET 8e Chapter 16 Form E
____ 7. Let where .
Which of the following equations does the line segment from to satisy?
Select the correct answer.
a.
b.
c. none of these
8. Use Green’s Theorem to find the work done by the force
in moving a particle in the positive direction once around the triangle with vertices , ,
and .
____ 9. A particle starts at the point , moves along the x-axis to (3, 0) and then along the semicircle
to the starting point. Use Green’s Theorem to find the work done on this particle by
the force field
Select the correct answer.
a.
b.
c. 0
d.
e.
10. 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 .
Stewart – Calculus ET 8e Chapter 16 Form E
____ 11. 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. Select the correct answer.
a. cannot be determined
b. zero
c. negative
d. positive
____ 12. Find the divergence of the vector field F.
Select the correct answer.
a.
b.
c.
d.
13. Find the curl of the vector field F.
Stewart – Calculus ET 8e Chapter 16 Form E
____ 14. The temperature at the point in a substance with conductivity is
Find the rate of heat flow inward across the cylindrical
Select the correct answer.
a.
b.
c.
d.
e.
15. 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.
Stewart – Calculus ET 8e Chapter 16 Form E
____ 16. Use the Divergence Theorem to calculate the surface integral ; that is, calculate the flux of
across .
S is the surface of the box bounded by the coordinate planes and the planes
.
Select the correct answer.
a.
b.
25
2
c.
d.
2
25
e.
17. Assuming that S satisfies the conditions of the Divergence Theorem and the scalar functions and
components of the vector fields have continuous second order partial derivatives, find
,
where a is the constant vector.
18. Find the area of the surface.
The part of the paraboloid ; ,
Stewart – Calculus ET 8e Chapter 16 Form E
____ 19. Find a parametric representation for the part of the elliptic paraboloid that lies in
front of the plane x = 0. Select the correct answer.
a.
b.
c.
d.
e.
20. Find the area of the part of the surface that lies between the planes x = 0, x = 4, ,
and z = 1.
Stewart – Calculus ET 8e Chapter 16 Form E
Answer Key
Stewart – Calculus ET 8e Chapter 16 Form F
____ 1. Find the gradient vector field of f. Select the correct answer.
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.)
3. Evaluate the line integral over the given curve C.
; ,
Stewart – Calculus ET 8e Chapter 16 Form F
____ 4. Find the work done by the force field F on a particle that moves along the curve C.
Select the correct answer.
; ,
a.
17
b.
19
c.
34
d.
71
2
5. Determine whether F is conservative. If so, find a function f such that .
6. 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
____ 7. 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
Select the correct answer.
a. 224
b. 135
c. 117
d. 70
Stewart – Calculus ET 8e Chapter 16 Form F
____ 8. Use Green’s Theorem to find the work done by the force
in moving a particle in the positive direction once around the triangle with vertices , ,
and .
Select the correct answer.
a. 7
b.
7
6
c.
7
3
d. 42
9. Find the curl of the vector field F.
____ 10. Find the curl of the vector field.
Select the correct answer.
a.
b.
c.
d.
e. None of these
Stewart – Calculus ET 8e Chapter 16 Form F
11. Find the value of the constant c such that the vector field
is the curl of some vector field F.
____ 12. Evaluate the surface integral. Round your answer to four decimal places.
S is surface
Select the correct answer.
a.
b.
c.
d.
e.
____ 13. Evaluate .
; S is the part of the torus with vector representation
, , .
Select the correct answer.
a. 0
b.
c.
d.
14. Find the mass of the surface S having the given mass density.
S is the hemisphere , ; the density at a point P on S is equal to the distance
between P and the xy-plane.