14.3 Change of Variables: Polar Coordinates
868
____ 3. Evaluate the double integral below.
a.
b.
c.
d.
e.
____ 4. Evaluate the double integral below.
a.
b.
c.
d.
e.
869
Chapter 14: Multiple Integration
____
5.
Identify the region of integration for the following integral.
a. d.
b. e.
c.
____ 6. Evaluate the following iterated integral by converting to polar coordinates.
14.3 Change of Variables: Polar Coordinates
870
a.
b.
c.
d.
e.
____ 7. Evaluate the iterated integral by converting to polar
coordinates.
a.
b.
c.
d.
e.
871 Chapter 14: Multiple Integration
____ 8.
Evaluate the following iterated integral by converting to polar coordinates.
a.
b.
c.
d.
e.
____ 9. Evaluate the iterated integral by converting to
polar coordinates. Round your answer to four decimal places.
10.5742
13.5742
17.5742
28.5742
14.5742
____ 10. Combine the sum of the two iterated integrals into a single integral by converting to polar
coordinates. Evaluate the resulting iterated integral.
a.
b.
c.
14.3 Change of Variables: Polar Coordinates
872
4
3
4
3
____ 11.
Given
use polar coordinates to set
.
a.
b.
c.
d.
e.
____ 12. Use a double integral in polar coordinates to find the volume of the solid in the first octant
bounded by the graphs of the equations given below.
a.
b.
c.
d.
873 Chapter 14: Multiple Integration
e.
____ 13. Use a double integral in polar coordinates to find the volume of the solid inside the
hemisphere
but outside the cylinder
.
a.
b.
c.
d.
e.
____ 14.
Find a such that the volume inside the hemisphere
and outside
the cylinder
is one-half the volume of the hemisphere. Round your answer to four
decimal places.
a.
b.
c.
d.
e.
14.3 Change of Variables: Polar Coordinates
874
____ 15.
Determine the diameter of a hole that is drilled vertically through the center of the
solid bounded by the graphs of the equations
if one-tenth
of the volume of the solid is removed. Round your answer to four decimal places.
1.2245
31.4490
7.2245
5.4490
15.2245
____ 16. Use a double integral to find the area of the shaded region as shown in the figure
below.
a.
b.
c.
d.
e.
875 Chapter 14: Multiple Integration
____ 17. Use a double integral to find the area enclosed by the graph of .
a.
b.
c.
d.
e.
____ 18. Use a double integral to find the area enclosed by the graph of .
14.3 Change of Variables: Polar Coordinates
876
a.
b.
c.
d.
e.
____ 19.
Use a double integral to find the area of the region inside the circle
and
outside the cardioid
. Round your answer to two decimal places.
a.
46.68
b.
58.34
c.
20.34
d.
55.34
e.
22.34
____ 20. Suppose the population density of a city is approximated by the model
where x and y are measured in miles. Integrate the
density function over the indicated circular region to approximate the population of the city.
Round your answer to the nearest integer.
417,127
417,029
833,901
833,903
416,951
877 Chapter 14: Multiple Integration
14.3 Change of Variables: Polar Coordinates
Answer Section
14.3 Change of Variables: Polar Coordinates
878
879 Chapter 14: Multiple Integration
14.4 Center of Mass and Moments of Inertia
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find the mass of the lamina described by the inequalities given that
its density is . (Hint: Some of the integrals are simpler in polar coordinates.)
768
128
512
256
392
____ 2. Find the mass of the lamina described by the inequalities
given that its density is . (Hint: Some of the integrals
are simpler in polar coordinates.)
a.
b.
c.
d.
e.
____ 3. Find the center of mass of the rectangular lamina with vertices and
for the density .
a.
b.
c.
d.
e.
14.4 Center of Mass and Moments of Inertia
880
____ 4.
Find the center of mass of the rectangular lamina with vertices
for the density
.
a.
b.
c.
d.
e.
____
5.
Find the mass of the triangular lamina with vertices
for
the density
.
a.
401k
b.
809k
c.
800k
d.
400k
e.
805k
____
6.
Find the mass of the triangular lamina with vertices
for
the density
.
139,968k
279,946k
139,958k
139,973k
279,936k
____ 7. Find the mass of the lamina bounded by the graphs of the equations
for the density .
a.
b.
881 Chapter 14: Multiple Integration
c.
d.
e.
____ 8. Find the center of mass of the lamina bounded by the graphs of the equations
for the density .
a.
b.
c.
d.
e.
____ 9. Find the center of mass of the lamina bounded by the graphs of the equations
for the density .
a.
b.
c.
d.
14.4 Center of Mass and Moments of Inertia 882
e.
____ 10. Find the mass of the lamina bounded by the graphs of the equations
for the density .
a.
b.
c.
d.
e.
____ 11. Find the center of mass of the lamina bounded by the graphs of the equations
for the density .
a.
b.
c.
d.
e.
883
Chapter 14: Multiple Integration
____
12.
Find the mass and center of mass of the lamina bounded by the graphs of the
equations given below for the given density.
a.
b.
c.
d.
e.
____ 13. Find the mass of the lamina bounded by the graphs of the equations
for the density .
a.
b.
c.
d.
e.
14.4 Center of Mass and Moments of Inertia
884
____ 14.
Set up and evaluate a double integral required to find the moment of inertia, I, about
the given line, of the lamina bounded by the graphs of the following equations. Use a computer
algebra system to evaluate the double integral.
a.
b.
c.
d.
e.
____ 15.
Set up the double integral required to find the moment of inertia I, about the line
of the lamina bounded by the graphs of the equations
for the density
. Use a computer algebra system to evaluate the double
integral. a.
b.
c.
d.
e.
885
Chapter 14: Multiple Integration
____
16.
Determine the location of the horizontal axis
for figure (b) at which a vertical
gate in a dam is to be hinged so that there is no moment causing rotation under the indicated loading
(see figure (a)). The model for is where is the y–coordinate of the centroid of the
gate, is the moment of inertia of the gate about the line , h is the depth of the centroid below
the surface, and A is the area of the gate.
a.
b.
c.
d.
e.
14.4 Center of Mass and Moments of Inertia
886
____ 17.
Determine the location of the horizontal axis
for figure (b) at which a vertical
gate in a dam is to be hinged so that there is no moment causing rotation under the indicated loading
(see figure (a)). The model for
is
where
is the y-coordinate of the centroid of the
gate, is the moment of inertia of the gate about the line , h is the depth of the centroid below
the surface, and A is the area of the gate.
a.
b.
c.
d.
e.
887 Chapter 14: Multiple Integration
14.4 Center of Mass and Moments of Inertia
Answer Section