686 Chapter 11: Vectors and the Geometry of Space
11.6 Surfaces in Space
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Identify the equation of the following graph.
a.
b.
c.
d.
e.
11.6 Surfaces in Space 687
____ 2. Identify the equation of the following graph.
a.
b.
c.
d.
e.
____ 3. Identify the following quadric surface.
a. hyperboloid of one sheet
b. elliptic cone
c. hyperboloid of two sheets
d. elliptic paraboloid
e. ellipsoid
688 Chapter 11: Vectors and the Geometry of Space
____ 4. Identify the following quadric surface.
a. hyperboloid of two sheets
b. elliptic cone
c. ellipsoid
d. elliptic paraboloid
e. hyperboloid of one sheet
____ 5. Identify the following quadric surface.
a. ellipsoid
b. elliptic cone
c. elliptic paraboloid
d. hyperboloid of one sheet
e. hyperboloid of two sheets
____ 6. Identify the following quadric surface.
a. elliptic paraboloid
b. elliptic cone
c. hyperboloid of two sheets
d. hyperboloid of one sheet
e. ellipsoid
____ 7. Identify the following quadric surface.
a. hyperboloid of one sheet
b. elliptic cone
c. elliptic paraboloid
d. hyperboloid of two sheets
e. hyperbolic paraboloid
11.6 Surfaces in Space 689
____ 8. Identify the following quadric surface.
a. Elliptic cone
b. Hyperboloid of two sheets
c. Elliptic paraboloid
d. ellipsoid
e. Hyperboloid of one sheet
____ 9. Find an equation of the surface of revolution generated by revolving the curve given
below in the indicated coordinate plane about the given axis.
Equation of Curve Coordinate Plane Axis of Revolution
a.
b.
c.
d.
e.
____ 10. Find an equation for the surface of revolution generated by revolving the curve
in the xz–plane about the z–axis.
a.
b.
c.
d.
e.
690 Chapter 11: Vectors and the Geometry of Space
____ 11. Given the equation of a surface of revolution find the equation of its
generating curve in the yz–plane rotating about the y–axis.
a.
b.
c.
d.
e.
____ 12. Use the shell method to find the volume of the solid below the surface formed by
revolving the curve about the z-axis and above the xy-plane.
a.
b.
c.
d.
e.
____ 13. Find the length of the minor axis of the ellipse generated when the surface
is intersected by the plane .
a.
b.
10 10
c.
55
d.
e.
____ 14. Find the equation of the surface satisfying the conditions, and identify the surface.
The set of points equidistant from the point (2, 4, 5 ) and the plane .
a.
b.
c.
d.
e.
11.6 Surfaces in Space 691
____ 15. Some planet is an oblate ellipsoid rather than a sphere because of the forces caused
by its rotation. The equatorial radius of this planet is 4961 miles and the polar radius is 4687 miles.
Find the equation of the ellipsoid. (Assume that the center of a planet is at the origin and that the trace
formed by the plane corresponds to the equator.)
a.
b.
c.
d.
e.
____ 16. The top of a rubber bushing designed to absorb vibrations in an automobile is the
surface of revolution generated by revolving the curve in the yz-plane about
the z-axis. Find an equation for the surface of revolution.
a.
b.
c.
d.
e.
____ 17. The top of a rubber bushing designed to absorb vibrations in an automobile is the
surface of revolution generated by revolving the curve in the yz-plane about
the z-axis. All measurements are in centimeters and the bushing is set on the xy-plane. Use the shell
method to find its volume. Round your answer to one decimal place.
a.
b.
c.
d.
e.
692 Chapter 11: Vectors and the Geometry of Space
____ 18. The top of a rubber bushing designed to absorb vibrations in an automobile is the
surface of revolution generated by revolving the curve in the yz-plane about
the z-axis. The bushing has a hole of 4 centimeters in diameter through its center and parallel to the
axis of revolution. All measurements are in centimeters and the bushing is set on the xy-plane. Find
the volume of the rubber bushing. Round your answer to two decimal places.
a.
b.
c.
d.
e.
11.6 Surfaces in Space 693
11.6 Surfaces in Space
Answer Section