§6.3: Volumes By Shells
Resources for the shell method for solids of revolution. The emphasis here is on choosing slices parallel to the axis of rotation, writing shell volume as $2\pi(\text{radius})(\text{height})(\text{thickness})$, deciding whether to integrate with respect to $x$ or $y$, and handling shells about the coordinate axes and about shifted lines such as $x=c$ or $y=c$.
Library, Handouts & Tools
| Resource Name | Type | Description | Link |
|---|---|---|---|
| OpenStax Calculus: Volumes of Revolution — Cylindrical Shells | Textbook Notes | Best single reference for the shell formula, the geometry of a typical shell, and setups about both the $y$-axis and the $x$-axis. | Read Notes |
| Paul's Notes: Method of Cylinders | Study Guide | Clear worked examples showing when shells are easier than washers, including problems about axes other than $x=0$ or $y=0$. | Read Notes |
| Paul's Practice Problems: Method of Cylinders | Practice Set | A focused shell-method problem set with answers, useful once the setup rules are in place and you want repetition. | Practice |
| CSUSM Worksheet: Volumes and Cylindrical Shells | Worksheet | A compact worksheet that mixes formula reminders with shell-method setup and evaluation practice. | View PDF |
| GeoGebra: Shell Method Interactive | Interactive | Dynamic visualization of a typical cylindrical shell, showing how radius, height, and thickness combine to build the volume integral. | Open Applet |
| MathDemos: Volumes of Solids of Revolution — Shell Method | Visual Guide | Good for intuition: nested shells, why the formula works, and how shell approximations fill out the solid. | Open Guide |
Video Lectures
| Topic | Source | Description | Link |
|---|---|---|---|
| Video Calculus Series (Shell Method entry) | Houston ACT | The UH Video Calculus index explicitly includes a cylindrical-shell-method entry, so this is the best Houston hub page to start from. | Open Page |
| Shell Method Calculus (Introduction) | Houston Math Prep | Introductory video that builds the shell formula from the geometry of a rotating rectangle. | Watch |
| Shell Method Calculus (Example 1) | Houston Math Prep | Worked example for setting up and evaluating a shell integral in a standard first-shells problem. | Watch |
| Houston Math Prep Calculus 2 Playlist | YouTube Playlist | Useful if you want the shell videos in the broader sequence of volume topics and adjacent lessons. | Open Playlist |
Focused Practice Path
| Skill | Best Resource | What to Focus On | Link |
|---|---|---|---|
| Build the formula | OpenStax | Start with why a shell contributes $2\pi(\text{radius})(\text{height})\,\Delta x$ or $2\pi(\text{radius})(\text{height})\,\Delta y$. | Start Here |
| Choose $dx$ or $dy$ correctly | Paul's Notes | Match strip direction to the axis of rotation: shell strips are parallel to the axis, not perpendicular to it. | Review |
| Radius and height from a picture | GeoGebra | Watch the shell move and identify which measurement is the radius and which is the shell height before writing the integral. | Open Applet |
| Practice standard setups | Paul's Practice | Do several clean shell setups in a row, especially around the $y$-axis and around shifted vertical lines like $x=c$. | Practice |
| Mixed worksheet review | CSUSM Worksheet | Finish with a worksheet-style set so you can move from graph reading to a complete shell integral efficiently. | View PDF |