This section introduces the shell method for computing volumes of solids of revolution.
| Skill | Description | Difficulty |
|---|---|---|
| Shell Method Formula | The cylindrical shell volume formula | Intermediate |
| Shell Method (y-axis) | Rotating about the y-axis | Intermediate |
| Shell Method (Other Axes) | Rotating about other vertical/horizontal lines | Advanced |
| Shells vs. Washers | Choosing the right method | Advanced |
When rotating a region about a vertical axis:
$$V = \int_a^b 2\pi x \cdot f(x)\,dx$$
where:
| Situation | Preferred Method |
|---|---|
| Rotating about y-axis, function of x | Shell method |
| Rotating about x-axis, function of x | Disk/Washer method |
| Region bounded by $x = g(y)$ | Consider both methods |
| One method gives simpler integral | Use the simpler one |
| ← Section 5.2 | Skills Index | Section 5.4 → |