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969
Chapter 15: Vector Analysis
d.
e.
____
11.
Find a vector-valued function whose graph is the ellipsoid
.
a.
b.
c.
d.
e.
____
12.
Find a vector-valued function whose graph is the part of the paraboloid
that lies inside the cylinder
.
a.
b.
c.
d.
e.
15.5 Parametric Surfaces
970
____ 13.
Write a set of parametric equations for the surface of revolution obtained by
revolving the graph of the function about the given axis.
a.
b.
c.
d.
e.
____ 14. Write a set of parametric equations for the surface of revolution obtained by
revolving the graph of the function about the x-axis.
a.
b.
c.
d.
e.
971
Chapter 15: Vector Analysis
____
15.
Find an equation of the tangent plane to the surface represented by the vector-valued
function at the given point.
a.
b.
c.
d.
e.
____ 16. Find the area of the surface given by , where
.
a.
b.
c.
d.
e.
____ 17. Find the area of the surface over the given region. Use a computer algebra system to verify
your results.
The sphere,
a.
b.
c.
d.
e.
15.5 Parametric Surfaces
972
____ 18.
Find the area of the surface over the given region. Use a computer algebra system to
verify your results.
The part of the cone,
where .
a.
b.
c.
d.
e.
____ 19. Find the area of the surface of revolu tion
.
a.
b.
c.
d.
e.
973 Chapter 15: Vector Analysis
____ 20. The surface of the dome on a new museum is given by
and is in
meters. Find the surface area of the dome.
a.
b.
c.
d.
e.
____ 21. Find a vector-valued function for the hyperboloid .
a.
b.
c.
d.
e.
____ 22. Determine the tangent plane for the hyperboloid .
a.
b.
c.
d.
e.
15.5 Parametric Surfaces
974
15.5 Parametric Surfaces
Answer Section
975
Chapter 15: Vector Analysis
15.6 Surface Integrals
976
15.6 Surface Integrals
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1.
Evaluate
, where
.
a.
b.
c.
d.
e.
____
2.
Evaluate
, where
.
a.
b.
c.
d.
e.
____
3.
Evaluate
, where S is
, first octant.
a.
b.
c.
d.
977
Chapter 15: Vector Analysis
____
4.
Use a computer algebra system to evaluate
where S is
. Round your answer to two decimal places.
124.82
1.07
62.41
141.01
64.00
____ 5. Use a computer algebra system to evaluate where S is
. Round your answer to two decimal places.
–475.02
–496.00
–203.00
–971.38
____ 6. Find the mass of the surface lamina S of density .
a.
b.
c.
d.
e.
____ 7. Evaluate , where
.
a.
b.
c.
d.
e.
15.6 Surface Integrals
978
____ 8.
Evaluate
, where
and S is given by
.
8
32
16
96
512
____ 9. Evaluate , where and S is given by
.
a. 10,000
b.
c.
d.
e.
____ 10. Evaluate , where and S is given by
.
a.
b.
c.
d.
e.
979 Chapter 15: Vector Analysis
____ 11. Evaluate , where
.
a.
b.
c.
d.
e.
____ 12. Find the flux of through S, , where is the upward unit normal
vector to S.
, first octant
a.
b.
c.
d.
e.
15.6 Surface Integrals
980
____ 13.
Find the flux of through S,
, where
is the upward unit normal
vector to S.
a.
b.
c.
d.
e.
____ 14. Find the flux of through
where N is the upward unit normal vector to S.
8
11
1
e. 4
____ 15. Find the flux of over the closed surface (let be the outward unit normal vector
of the surface).
S: cube bounded by
200
800
400
1,000
600
981 Chapter 15: Vector Analysis
____ 16. Let be an electrostatic field. Use Gauss's Law to find the total
charge enclosed by the closed surface consisting of the hemisphere and its circular
base in the xy-plane.
a.
b.
c.
d.
e.
____ 17. Find for the lamina with uniform density of 1. Use a
computer algebra system to verify your result.
a.
b.
c.
d.
e.
____ 18. Let S be the surface oriented upwards. Use a computer
algebra system to find the rate of mass flow of a fluid of density through S if the velocity field is
given by .
a.
b.
c.
d.
e.
15.6 Surface Integrals
982
15.6 Surface Integrals
Answer Section
15.7 Divergence Theorem
983
15.7 Divergence Theorem
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Use the Divergence Theorem to evaluate Verify your answer by
evaluating the integral as a triple integral.
S: cube bounded by the planes
a.
b.
c.
d.
e.
____ 2.
Let
and let S be the cylinder
Verify the Divergence Theorem by evaluating
as a surface integral and as a triple
integral.
a.
b.
c.
d.
e.
