Cylinders
Textbook Reference
| Primary source | OpenStax Calculus Volume 3, Section 2.6: “Quadric Surfaces” |
| Direct link | https://openstax.org/books/calculus-volume-3/pages/2-6-quadric-surfaces |
| Textbook used in class | Stewart, Calculus, Section 12.6: “Cylinders and Quadric Surfaces” (Examples 1, 2) |
Opening Scenario
A soda can is a circular cylinder: every horizontal cross-section is a circle of the same radius. In mathematics, a cylinder is any surface generated by taking a plane curve and sweeping it in a direction perpendicular to the plane containing the curve. The equation is simple to recognize: one variable is absent.
Quick Reference
Definition. A cylinder is a surface consisting of all parallel lines that intersect a plane curve (the directrix) and are parallel to a fixed direction.
Recognition rule. In an equation in $x$, $y$, $z$:
- If $z$ is absent: the surface is a cylinder whose cross-sections perpendicular to the $z$-axis all look the same.
- If $y$ is absent: the surface is a cylinder extending in the $y$-direction.
- If $x$ is absent: extending in the $x$-direction.
Key Concepts
1. Circular Cylinders
$x^2 + y^2 = r^2$ describes a circular cylinder of radius $r$ extending along the $z$-axis. Every horizontal slice ($z = c$) is a circle of radius $r$.
2. Parabolic and Other Cylinders
$z = y^2$ is a parabolic cylinder: the cross-section in the $yz$-plane is the parabola $z = y^2$, and this parabola is swept in the $x$-direction.
Example 1. Sketch and describe $z = \sin y$. (Stewart 12.6, Example 1.)
The $y$-$z$ cross-section is the sine curve $z = \sin y$. This is swept in the $x$-direction to produce a surface that looks like corrugated sheet metal.
thinking $x^2 + y^2 = 4$ in 3D is a circle. In 2D, $x^2 + y^2 = 4$ is a circle. In 3D, it is an infinite circular cylinder of radius $2$ centered on the $z$-axis, because $z$ is unrestricted. The curve becomes a surface when the missing variable extends the shape into the third dimension.
Practice
Problem 1. Describe the surface $x^2 + z^2 = 9$ in 3D.
Show answer
$y$ is absent, so the surface extends in the $y$-direction. In the $xz$-plane, $x^2 + z^2 = 9$ is a circle of radius $3$. The surface is a circular cylinder of radius $3$ centered on the $y$-axis.
Common Misconceptions
an equation missing one variable in 3D still describes a bounded curve.
This is the iconic-graph error. When one variable is absent from an equation, the surface extends infinitely in that variable’s direction. The equation $x^2 + y^2 = 4$ in 3D is not the circle of radius 2; it is an infinite circular cylinder whose axis is the $z$-axis, because $z$ is completely free. The circle $x^2 + y^2 = 4$ in 3D requires the additional constraint $z = 0$ (or any fixed value) to become a curve.