Objectives
- To understand the correspondence between equations missing either \(x\text{,}\) \(y\text{,}\) or \(z\) and a cylinder surface
- To understand how a generating curve and direction to stretch creates a cylinder surface
Section 3.5 Cylinder Surfaces
Activity 3.5.1. Preview Activity.
(a)
Draw each of the following sets in 3D:
- \(\displaystyle \{(3,y,0)| y \text{ is real}\}\)
- \(\displaystyle \{(3,y,z)| y,z \text{ are real}\}\)
- How do these graphs compare to the plot of \(x=3\) in space?
- \(\displaystyle \{(0,y,z)| y+z=4 \}\)
- \(\displaystyle \{(x,y,z)| y+z=4 \}\)
- How do these graphs compare to the plot of \(y+z=4\) in space?
(b)
In each of the following problems draw in three-dimensional space and describe the intersection of set \(S\) with the given axis. There is no need to know the graph of S to do this.
- \(S=\{(x,y,z)|z=x^2+xy^2\}\) with the \(y\)-axis (Your answer is contained in the y-axis
- \(S=\{(x,y,z)|z=x^2+(2+y)^3 x +y^2\}\) with the \(x\)-axis
- \(S=\{(x,y,z)|z=x \sin(y)\}\) with the \(z\)-axis
Subsection 3.5.1 CalcVR Lesson on Cylinder Surfaces
This lesson begins with looking at the example \(y=|x|\) as a surface in space. By looking at the intersections with different fundamental planes with constant values of \(z\text{,}\) we are able to construct the cylinder surface and see an example of a generating curve. Several other examples of cylinder surfaces are examined in terms of the graph or the algebraic form of the surface. This lesson finishes with questions about identifying cylinder surfaces based on the surface plot only.
Activity 3.5.2. Drawing Cylinder Surfaces by Hand.
In this problem the graph of \(x=9-z^2\) will be drawn in three-dimensional space. The graph of \(x=9-z^2\) consists of all points in the set \(\{(x,y,z)|x=9-z^2\}\text{.}\)
(a)
Draw in three-dimensional space all points where the plane \(y=0\) intersects the graph of \(x=9-z^2\text{.}\)
(b)
Draw in three-dimensional space all points where the plane \(y=1\) intersects the graph of \(x=9-z^2\text{.}\)
(c)
Draw in three-dimensional space all points where the plane \(y=-1\) intersects the graph of \(x=9-z^2\text{.}\)
(d)
What happens as \(y\) is given more and more positive and negative values?
(e)
Draw the graph of \(x=9-z^2\) in space.
Activity 3.5.3. Drawing Cylinder Surfaces by Hand.
In each of the following cases, consider the set of all points in three-dimensional space that satisfy the given equation. Draw it in three-dimensional space and be sure to draw the generating curve for each surface.
- \(\displaystyle x=2z\)
- \(\displaystyle y=\sin(z)\)
- \(\displaystyle x^2-z^2=1\)
Activity 3.5.4.
Reflect on what was done in problems from the CalcVR lesson on Cylindrical Coordinates. How, in general, is the graph of an equation where only two variables appear drawn in three-dimensional space? Try to be specific in your explaination for how to draw these types of graphs.
Subsection 3.5.2 Comments to Instructors
The lesson on Cylindrical Surfaces introduces several functionalities in the CalcVR app, including the ability to draw on a 2D canvas and have your drawing plotted appropriately in 3D space.
