Stewart – Calculus ET 8e Chapter 16 Form F
____ 15. Evaluate the surface integral where S is the surface with parametric equations ,
.
Select the correct answer.
a.
b.
c.
d.
e.
16. Use Stokes’ Theorem to evaluate .
;
S is the part of the paraboloid lying below the plane and oriented with normal
pointing downward.
____ 17. Use Stoke’s theorem to evaluate where and C is the
boundary of the part of the plane in the first octant. Select the correct answer.
a. 69
b. 16
c. 49
d. 0
e. 23
18. 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
.
Stewart – Calculus ET 8e Chapter 16 Form F
____ 19. Match the equation with one of the graphs below.
a.
c.
b. d.
____ 20. Find the area of the part of the surface that lies between the planes x = 0, x = 4, ,
and z = 1. Select the correct answer.
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 16 Form F
Answer Key
Stewart – Calculus ET 8e Chapter 16 Form G
____ 1. Which plot illustrates the vector field
a.
c.
b. d.
2. Find the gradient vector field of f.
3. 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.)
4. Find the exact mass of a thin wire in the shape of the helix
if the density is 4.
Stewart – Calculus ET 8e Chapter 16 Form G
____ 5. Find the work done by the force field on a particle that moves along the parabola
Select the correct answer.
a.
b.
c.
d.
e.
6. Evaluate the line integral over the given curve C.
; ,
Stewart – Calculus ET 8e Chapter 16 Form G
____ 7. Find the work done by the force field F on a particle that moves along the curve C.
Select the correct answer.
; ,
a.
85
6
b.
3
c.
13
2
d.
13
____ 8. Determine whether F is conservative. If so, find a function f such that .
Select the correct answer.
a.
b. not conservative
c.
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 .
Stewart – Calculus ET 8e Chapter 16 Form G
____ 10. Let
Select the correct answer.
a. 6
b. 12
c. 18
d. 30
e. None of these
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.
____ 12. Suppose that where g is a function of one variable such that
.
Evaluate where S is the sphere
Select the correct answer.
a.
b.
c.
d.
e. None of these
Stewart – Calculus ET 8e Chapter 16 Form G
13. Evaluate the surface integral.
S is the part of the plane that lies in the first octant.
14. Evaluate the surface integral. Round your answer to four decimal places.
S is surface
15. Evaluate , that is, find the flux of F across S.
; S is the hemisphere ; n points upward.
____ 16. 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. Select the correct answer.
a.
b.
c.
d.
e. none of these
Stewart – Calculus ET 8e Chapter 16 Form G
17. Use Stokes’ Theorem to evaluate .
;
S is the part of the paraboloid lying below the plane and oriented with normal
pointing downward.
18. Use Stoke’s theorem to evaluate where and C is the
boundary of the part of the plane in the first octant.
____ 19. 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
Select the correct answer.
a.
b.
c.
d.
e.
20. 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.
Stewart – Calculus ET 8e Chapter 16 Form G
Answer Key
Stewart – Calculus ET 8e Chapter 16 Form H
____ 1. Match the vector field with its plot.
a.
c.
b.
d.
2. Find the work done by the force field on a particle that moves along the parabola
Stewart – Calculus ET 8e Chapter 16 Form H
3. Find the work done by the force field F on a particle that moves along the curve C.
; ,
____ 4. 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.
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 this result to evaluate
, where C is any path from to .
; and
Select the correct answer.
a.
b.
c.
d.
Stewart – Calculus ET 8e Chapter 16 Form H
8. 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, ).
____ 9. 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 .
Select the correct answer.
a.
b.
c.
d.
e.
10. Let
Stewart – Calculus ET 8e Chapter 16 Form H
____ 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.
a. zero
b. cannot be determined
c. negative
d. positive
____ 12. Evaluate the surface integral.
S is the part of the plane that lies in the first octant. Select the correct answer.
a.
b.
c.
d.
e.
13. 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.
Stewart – Calculus ET 8e Chapter 16 Form H
____ 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. Select the correct answer.
a.
b.
c. 64
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.
16. Evaluate the surface integral where S is the surface with parametric equations ,
.
____ 17. , where S consists of the hemisphere and the disk
in the -plane. Select the correct answer.
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.
Stewart – Calculus ET 8e Chapter 16 Form H
____ 19. Match the equation with one of the graphs below.
a.
c.
b. d.
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 H
Answer Key