14.6 Triple Integrals and Applications
908
____ 18. Find the average value of
tetrahedron in the first octant with vertices
over the region Q, where Q is a and
. The average
value of a continuous function over a solid region Q is , where V is
the volume of the solid region Q.
a.
b.
c.
d.
e.
909 Chapter 14: Multiple Integration
14.6 Triple Integrals and Applications
Answer Section
14.7 Triple Integrals in Other Coordinates
910
14.7 Triple Integrals in Other Coordinates
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Evaluate the following iterated integral.
a.
b.
c.
d.
e.
____ 2. Evaluate the following iterated integral.
a.
b.
c.
d.
e.
911 Chapter 14: Multiple Integration
___ 3. Evaluate the iterated integral .
a.
b.
c.
d.
e.
____ 4. Evaluate the following iterated integral.
224
3
224
3
288
7
40
288
7
14.7 Triple Integrals in Other Coordinates
912
____ 5.
Convert the integral below from rectangular coordinates to both cylindrical and
spherical coordinates, and evaluate the simpler iterated integral.
a.
b.
c.
d.
e.
____ 6. Convert the integral below from rectangular coordinates to both cylindrical and
spherical coordinates, and evaluate the simpler iterated integral.
a.
b.
c.
d.
e.
913
Chapter 14: Multiple Integration
____
7.
Use cylindrical coordinates to find the volume of the solid inside both
and
.
a.
b.
c.
d.
e.
____ 8. Use cylindrical coordinates to find the volume of the solid bounded above by
and below by .
a.
b.
c.
d.
e.
14.7 Triple Integrals in Other Coordinates
914
____ 9.
Use cylindrical coordinates to find the volume of the solid inside the sphere
and above the upper nappe of the cone .
a.
b.
c.
d.
e.
____ 10. Use cylindrical coordinates to find the mass of the solid where
.
a.
b.
c.
d.
e.
915
Chapter 14: Multiple Integration
____
11.
Use cylindrical coordinates to find the volume of the cone
where
and
.
a.
b.
c.
d.
e.
____ 12. Use spherical coordinates to find the volume of the solid inside and
outside , and above the xy-plane.
a.
b.
c.
d.
e.
14.7 Triple Integrals in Other Coordinates
916
____ 13. Use spherical coordinates to find the volume of the solid inside the torus given by
.
a.
b.
c.
d.
e.
____ 14. Use spherical coordinates to find the volume of the solid between the spheres
and and inside the cone .
a.
b.
c.
d.
e.
____ 15. Use spherical coordinates to find the mass of the sphere with the given density. The density
at any point is proportional to the distance of the point from the z-axis.
a.
b.
c.
d.
e.
917
Chapter 14: Multiple Integration
____
16.
Use spherical coordinates to find the z coordinate of the center of mass of the solid
lying between two concentric hemispheres of radii 4 and 7, and having uniform density k.
a.
b.
c.
d.
e.
____ 17. Use spherical coordinates to find the z coordinate of the center of mass of the solid lying
between two concentric hemispheres of radii 6 and 7, and having uniform density k.
a.
b.
c.
d.
e.
14.7 Triple Integrals in Other Coordinates
918
14.7 Triple Integrals in Other Coordinates
Answer Section