MTH 227 Handout – Section 11.6 (Surfaces in Space)
Review the first 5 sections of chapter 11 to refresh your memory on vector calculus
Section 11.2 and 11.5 introduced 2 special types of surfaces in space (spheres and planes – demonstrate on
board or visually). Other types of surfaces are explored in this section. First, consider a cylindrical surface.
2 2 2
Consider a circle in the xy-plane (x + y = a ) Now consider the surface formed when you take a
line perpendicular to this circle (ruling) and rotate it
- generating curve or directrix -
around this curve
y
z z
x y y
x x
(note: you could also visualize this cylinder as a
circle extended up and down the z-axis)
What would be the equation of this cylinder?
2 2 2
(hint - you can generate any parallel ruling by fixing x and y coordinates to x + y = a and allowing z to be all real values)
Examples: Sketch the surface represented by each equation…
2
y=3 y=x z = sin y 0 ≤ y ≤ 2π
nd 2 2 2
The equation of a quadric surface in space is a 2 degree equation in 3 variables Ax + By + Cz + Dxy +…
There are 6 basic types of quadric surfaces (review these in class on pages 814-815 and review standard
equations below)
To classify a quadric surface, (1) write the surface in standard form and (2) determine some of the traces…
Examples: Classify and sketch the surface given by…
xy-trace (z = 0): Standard form:
xz-trace (y = 0): xy-trace (z = 0):
yz-trace (x = 0): xz-trace (y = 0):
yz-trace (x = 0):
Homework Exercises: # 1-4, 7, 8, 9, 11, 15, 20, 21, 47, 49, 50 Challenge: # 32, 52
Standard equations
xy-trace (z = 0): Circle:
xz-trace (y = 0): Ellipse: + =1
yz-trace (x = 0):
Hyperbola: – =1
Another type of surface considered is a surface of revolution (section 7.4 reviews a method for finding the
area of such a surface – disk / washer method). Look a graph of radius function y = r(z) in the yz-plane (point /
line / curve) and the effect when rotating this graph around the z-axis (pg 818).
z Examining these graphs along with the graph from pg.
818, we see that the cross sections are circles…
If a radius function is revolved around a coordinate
y
axes, the equation of the resulting surface of revolution
x has one of the following forms
1. Revolved around x-axis:
2. Revolved around y-axis:
3. Revolved around z-axis:
Examples: Find an equation for the surface of revolution formed by revolving the following graphs
z = 3y around the y-axis xy = 2 around the x-axis
We can reverse this process to find a generating curve and axis….
Example: Find a generating curve and the axis of revolution given by
Homework Exercises: # 1-4, 7, 8, 9, 11, 15, 20, 21, 47, 49, 50 Challenge: # 32, 52