11.6 Cylinders and Quadric Surfaces by SGLee - HSKim - JHLee
1.(a) What does the equation 
represent as a curve in 
?
Equation 
represents a parabola of slope 
passing through origin in 
.
(b) What does it represent as a surface in 
?
The equation of the graph is 
, which doesn't involve 
in 
.
This means that any vertical plane with equation 
(parallel to the 
plane) intersects the graph in a curve with 
,
that is, a parabola. Below figure shows how the graph is formed by taking the parabola 
in the 
plane
and moving it in the direction of the 
axis.
So the graph is a surface, called a parabolic cylinder, made up of infinitely many shifted copies of the same parabola.
2. (a) Sketch the graph of 
as a curve in 
.
as a surface in 
.
3. 
4. 
5. 
, 
, 
. Then, identify the surface and sketch it.
6. 
are ellipses of the form 
, 
,
are parabolas of the form 
,
are parabolas of the form 
.
8. 
are hyperbolas of the form 
,
are circles of the form 
, 
,
are hyperbolas of the form 
.
10. 
,
and axis the horizontal line 
.
12. 
and 
gives 
or
, a hyperboloid of one sheet.
14. 
15. 
16. Sketch the region bounded by the surfaces 
and 
for 
.
about the 
-axis.
for which
to the 
-axis is twice the distance from 
to the 
-plane. Identify the surface.
be an arbitrary point whose distance from th 
axis is twice its distance from the 
plane.
to the 
axis is 
and the distance from 
to the 
plane(
) is 
.
.
axis.
