Triple Integrals in Cylindrical
Triple Integrals in 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” (Examples 1, 2) |
Quick Reference
Volume element: $dV = r\,dr\,d\theta\,dz$.
Triple integral in cylindrical: $$\iiint_E f(x,y,z)\,dV = \iiint_{E'} f(r\cos\theta, r\sin\theta, z)\,r\,dr\,d\theta\,dz.$$
Typical order: integrate $z$ first (top to bottom of solid), then $r$ and $\theta$ over the projection.
Motivation
When the solid has circular symmetry around the $z$-axis (a cylinder, cone, paraboloid, or sphere), cylindrical coordinates dramatically simplify the limits and the integrand. The $x^2+y^2$ terms in the Cartesian description become $r^2$, and circular boundaries become constant-$r$ limits.
The factor $r$ in $dV = r\,dr\,d\theta\,dz$ is essential and comes from the Jacobian of the cylindrical transformation -- the same $r$ that appears in polar $dA = r\,dr\,d\theta$.
Key Concept
Setup procedure:
- Describe the solid in cylindrical: find $r$-limits, $\theta$-limits, and $z$-limits (as functions of $r$ and $\theta$ if necessary).
- Replace $x = r\cos\theta$, $y = r\sin\theta$ in the integrand.
- Include the factor $r$ in $dV$.
- Integrate in the chosen order.
Worked Examples
Example 1. Find the volume of the solid bounded by the paraboloid $z = x^2+y^2$ and the plane $z = 4$ (same as the previous skill, now in cylindrical). (Stewart 15.8, Example 1.)
In cylindrical: $z = r^2$ (paraboloid), $z = 4$ (plane), $0 \leq r \leq 2$ (intersection at $r = 2$), $0 \leq \theta \leq 2\pi$.
$$V = \int_0^{2\pi}\int_0^2\int_{r^2}^4 r\,dz\,dr\,d\theta = \int_0^{2\pi}\int_0^2 r(4-r^2)\,dr\,d\theta.$$
$$= 2\pi\int_0^2(4r-r^3)\,dr = 2\pi\left[2r^2-\frac{r^4}{4}\right]_0^2 = 2\pi(8-4) = 8\pi.$$
Example 2. Evaluate $\iiint_E \sqrt{x^2+y^2}\,dV$ where $E$ is the solid inside the cylinder $x^2+y^2 = 1$, above $z=0$, and below $z = 2$. (Stewart 15.8, Example 2.)
In cylindrical: $\sqrt{x^2+y^2} = r$. Limits: $0 \leq r \leq 1$, $0 \leq \theta \leq 2\pi$, $0 \leq z \leq 2$.
$$\int_0^{2\pi}\int_0^1\int_0^2 r\cdot r\,dz\,dr\,d\theta = \int_0^{2\pi}\int_0^1 2r^2\,dr\,d\theta = 2\pi\cdot 2\cdot\frac{1}{3} = \frac{4\pi}{3}.$$
$dV = dr\,d\theta\,dz$ in cylindrical coordinates. The volume element in cylindrical coordinates is $dV = r\,dr\,d\theta\,dz$, not $dr\,d\theta\,dz$. The factor $r$ is a Jacobian factor that accounts for the fact that cylindrical “cells” at distance $r$ from the $z$-axis have a circumference of $2\pi r$, making them larger in the $\theta$-direction than cells near the origin.
Leveled Practice
Problem 1. Use cylindrical coordinates to compute $\iiint_E z\,dV$ where $E$ is the solid bounded by the cone $z = r$ and the cylinder $r = 3$, with $0 \leq z \leq 3$.
Show answer
Limits: $0 \leq \theta \leq 2\pi$, $0 \leq r \leq 3$, $r \leq z \leq 3$.
$$\int_0^{2\pi}\int_0^3\int_r^3 z\cdot r\,dz\,dr\,d\theta = 2\pi\int_0^3 r\left[\frac{z^2}{2}\right]_r^3 dr = 2\pi\int_0^3 r\cdot\frac{9-r^2}{2}\,dr.$$
$$= \pi\int_0^3(9r-r^3)\,dr = \pi\left[\frac{9r^2}{2}-\frac{r^4}{4}\right]_0^3 = \pi\left(\frac{81}{2}-\frac{81}{4}\right) = \frac{81\pi}{4}.$$