890 Chapter 14: Multiple Integration
14.8 Change of Variables: Jacobians
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find the Jacobian for the following change of variables:
a.
b.
c.
d.
e.
____ 2. Find the Jacobian for the change of variables given below.
a.
b.
c.
d.
e.
14.8 Change of Variables: Jacobians 891
____ 3. Find the Jacobian for the indicated change of variables.
a.
b.
c.
d.
e.
____ 4. Find the Jacobian for the following change of variables:
a.
b.
c.
d.
e.
892 Chapter 14: Multiple Integration
____ 5. Find the Jacobian for the following change of variables:
a.
b.
c.
d.
e.
____ 6. Sketch the image S in the uv-plane of the region R in the xy-plane using the given
transformation.
14.8 Change of Variables: Jacobians 893
a. d.
b. e.
c.
894 Chapter 14: Multiple Integration
____ 7. Use the indicated change of variables to evaluate the following double integral.
a.
b.
c.
d.
e.
____ 8. Use the indicated change of variables to evaluate the following double integral.
a.
b.
14.8 Change of Variables: Jacobians 895
c.
d.
e.
____ 9. Use the following change of variables to evaluate the double integral .
Round your answer to four decimal places.
a.
b.
c.
d.
e.
____ 10. Use a change of variables to find the volume of the solid region lying below the
surface and above the plane region R: region bounded by the square with vertices
.
a.
b.
c.
d.
e.
896 Chapter 14: Multiple Integration
____ 11. Use a change of variables to find the volume of the solid region lying below the
surface and above the plane region R: region bounded by the parallelogram
with vertices . Round your answer to two decimal places.
a.
b.
c.
d.
e.
____ 12. Use a change of variables to find the volume of the solid region lying below the
surface and above the plane region R: region bounded by the parallelogram
with vertices . Round your answer to two decimal places.
a.
b.
c.
d.
e.
____ 13. Use a change of variables to find the volume of the solid region lying below the
surface and above the plane region R: region bounded by the graphs of
(Hint: Let .) Round your answer to two decimal places.
a.
b.
c.
d.
e.
14.8 Change of Variables: Jacobians 897
____ 14. Find a transformation that when applied to
region , its image will be .
a.
b.
c.
d.
e.
____ 15. Consider the region in the xy-plane bounded by the ellipse and the
transformation and . Find .
a.
b.
c.
d.
e.
898 Chapter 14: Multiple Integration
____ 16. Consider the region R in the xy-plane bounded by the ellipse and the
transformation and . Find the area of the ellipse. Round your answer to two decimal
places.
a.
b.
c.
d.
e.
____ 17. If and then the Jacobian of x, y, and z
with
respect to u, v, and w is .
Find the Jacobian for the following change of variables:
a.
b.
c.
d.
e.
____ 18. If and then the Jacobian of x, y, and z
with respect to u, v, and w is .
14.8 Change of Variables: Jacobians 899
Find the Jacobian for the following change of variables:
a.
b.
c.
d.
e.
900 Chapter 14: Multiple Integration
14.8 Change of Variables: Jacobians
Answer Section
MULTIPLE CHOICE