Cylindrical Coordinates
Textbook Reference
| Primary source | OpenStax Calculus Volume 3, Section 5.5: “Triple Integrals in Cylindrical and Spherical Coordinates” |
| Direct link | https://openstax.org/books/calculus-volume-3/pages/5-5-triple-integrals-in-cylindrical-and-spherical-coordinates |
| Textbook used in class | Stewart, Calculus, Section 15.8: “Triple Integrals in Cylindrical Coordinates” |
Quick Reference
Cylindrical coordinates $(r, \theta, z)$: $$x = r\cos\theta, \quad y = r\sin\theta, \quad z = z.$$ $$r^2 = x^2+y^2, \quad \tan\theta = y/x.$$
| Surface | Cylindrical equation |
|---|---|
| Cylinder $x^2+y^2 = a^2$ | $r = a$ |
| Half-plane from $z$-axis | $\theta = \theta_0$ |
| Cone $z = \sqrt{x^2+y^2}$ | $z = r$ |
| Paraboloid $z = x^2+y^2$ | $z = r^2$ |
Motivation
Cylindrical coordinates are polar coordinates for the $xy$-plane, extended to 3D by keeping $z$ unchanged. They are natural for problems involving cylinders, cones, and paraboloids -- surfaces with circular cross-sections. The $z$-axis is the axis of symmetry; $r$ measures radial distance from it; $\theta$ measures the angle around it.
Key Concept
Cylindrical coordinates are a direct hybrid: polar $(r,\theta)$ for the horizontal plane and Cartesian $z$ for height. Conversion is the same as polar in the $xy$-plane: $x = r\cos\theta$, $y = r\sin\theta$, $z = z$. The expression $x^2+y^2 = r^2$ simplifies many integrands and surfaces.
Worked Examples
Example 1. Express the equation $x^2+y^2+z^2 = 4$ in cylindrical coordinates.
$r^2 + z^2 = 4$.
This is the sphere of radius 2 centered at the origin. In cylindrical form, it shows the interplay between the radial distance from the $z$-axis ($r$) and the height ($z$).
Example 2. Convert the point $(r,\theta,z) = (2, \pi/3, 5)$ to Cartesian.
$x = 2\cos(\pi/3) = 2\cdot\frac{1}{2} = 1$, $y = 2\sin(\pi/3) = 2\cdot\frac{\sqrt{3}}{2} = \sqrt{3}$, $z = 5$.
Cartesian: $(1, \sqrt{3}, 5)$.
cylindrical coordinates have three dimensions, so $\theta$ ranges over more than $2\pi$. The angle $\theta$ always ranges over an interval of total length $2\pi$ (for a full revolution), just as in polar coordinates. The $z$-coordinate is independent and ranges over whatever interval is needed for the solid. Cylindrical coordinates are not spherical coordinates -- $r$ is NOT a distance from the origin but a distance from the $z$-axis.
Leveled Practice
Problem 1. Describe the solid $1 \leq r \leq 2$, $0 \leq z \leq 3$ in words.
Show answer
This is a cylindrical shell (a hollow cylinder): the region between the cylinders $x^2+y^2 = 1$ and $x^2+y^2 = 4$, from height $z = 0$ to $z = 3$. Its volume is $\pi(4-1)(3) = 9\pi$.