Shell Method: Non-Standard Axes of Rotation
Quick Reference
Axis of Rotation Radius Strip Direction Integrate $y$-axis ($x = 0$) $x$ Vertical $dx$ $x = k$ (vertical) $\lvert x - k \rvert$ Vertical $dx$ $x$-axis ($y = 0$) $y$ Horizontal $dy$ $y = k$ (horizontal) $\lvert y - k \rvert$ Horizontal $dy$ Universal rule: Radius = distance from shell to axis
Before You Start
1. Can you set up a shell integral for rotation about the $y$-axis?
For $y = f(x)$ rotated about the $y$-axis: $$V = \int_a^b 2\pi x f(x) \, dx$$
If this is unclear, master Shell Method: y-axis first.
2. What is the distance from $x = 3$ to the line $x = 7$?
Answer: $\vert 3 - 7\vert = \vert -4\vert = 4$
The distance between a point and a vertical line is always positive. This concept is essential for finding shell radii.
3. Can you solve for $x$ in terms of $y$ when given $y = \sqrt{x}$?
Answer: Square both sides: $y^2 = x$, so $x = y^2$.
For rotation about horizontal axes, we often need to express boundaries as functions of $y$.
Beyond the y-axis
Real problems don’t always rotate about the $y$-axis. The axis might be $x = 5$, $x = -2$, or even a horizontal line like $y = 3$. The shell method adapts to all these cases with one key insight:
The radius is always the distance from the shell to the axis of rotation.
Once you internalize this rule, any axis becomes manageable.
Prerequisite Map
Quick Reference
| Property | Value |
|---|---|
| Chapter | 5 - Applications of Integration |
| Section | 5.3 |
| Difficulty | Intermediate |
| Time | ~25 minutes |
Key Concepts
The Universal Rule
For any axis of rotation:
$$\text{Radius} = \text{distance from shell to axis}$$
This is always positive. Use absolute value when the region might span both sides of the axis.
Case 1: Vertical Axis $x = k$
When rotating about the vertical line $x = k$:
Step 1: Use vertical strips (integrate with $dx$).
Step 2: Find the radius = distance from strip at $x$ to the line $x = k$:
$$\text{radius} = \lvert x - k \rvert$$
Step 3: Simplify based on where the region lies:
| Region Position | Radius Formula |
|---|---|
| Entirely right of axis ($x > k$ for all $x$) | $x - k$ |
| Entirely left of axis ($x < k$ for all $x$) | $k - x$ |
| Spans across axis | Split integral or use $\lvert x - k \rvert$ |
Step 4: Set up the integral: $$V = \int_a^b 2\pi \cdot (\text{radius}) \cdot (\text{height}) \, dx$$
Visualizing Vertical Axes
Axis x = k
│
│
←─────────┼──────────→
k - x │ x - k
(radius │ (radius
if x < k) │ if x > k)
│
Example: For axis x = 2 and strip at x = 5:
radius = 5 - 2 = 3
Example: For axis x = 2 and strip at x = -1:
radius = 2 - (-1) = 3Case 2: Horizontal Axis $y = k$
When rotating about the horizontal line $y = k$:
Step 1: Use horizontal strips (integrate with $dy$).
Step 2: Express boundaries as functions of $y$ (solve $y = f(x)$ for $x$).
Step 3: Find the radius = distance from strip at height $y$ to the line $y = k$:
$$\text{radius} = \lvert y - k \rvert$$
Step 4: Identify the “width” of each strip: $$\text{width} = (\text{right boundary}) - (\text{left boundary})$$
Step 5: Set up the integral: $$V = \int_c^d 2\pi \cdot (\text{radius}) \cdot (\text{width}) \, dy$$
Case 3: Rotation About the $x$-axis
This is the special case of $y = k$ with $k = 0$:
$$V = \int_c^d 2\pi y \cdot (\text{width}) \, dy$$
The radius is simply $y$ (the height of the strip).
Summary Table
| Axis | Strip Direction | Variable | Radius | Typical Setup |
|---|---|---|---|---|
| $y$-axis ($x = 0$) | Vertical | $dx$ | $x$ | $\int 2\pi x f(x) \, dx$ |
| $x = k$ | Vertical | $dx$ | $\lvert x - k \rvert$ | $\int 2\pi \lvert x-k \rvert f(x) \, dx$ |
| $x$-axis ($y = 0$) | Horizontal | $dy$ | $y$ | $\int 2\pi y \cdot \text{width} \, dy$ |
| $y = k$ | Horizontal | $dy$ | $\lvert y - k \rvert$ | $\int 2\pi \lvert y-k \rvert \cdot \text{width} \, dy$ |
📜 Why These Cases Cover Everything
Any line in the plane is either vertical ($x = k$) or has some other slope. For volumes of revolution in calculus courses, we only rotate about vertical or horizontal axes because:
- The resulting solid has circular cross-sections
- The integrals remain tractable
Rotation about slanted axes (like $y = x$) is possible but requires more advanced techniques (typically covered in multivariable calculus using different coordinate systems).
Practice Problems
For each situation, determine the radius of a shell at position $x$:
- Rotating about $x = 4$, with the region in $[0, 3]$
- Rotating about $x = -2$, with the region in $[0, 3]$
- Rotating about $x = 1$, with the region in $[2, 5]$
Find the volume of the solid obtained by rotating the region bounded by $y = x - x^2$ and $y = 0$ about the line $x = 2$.
Use cylindrical shells to find the volume of the solid obtained by rotating the region bounded by $y = \sqrt{x}$, $y = 0$, and $x = 4$ about the $x$-axis.
Find the volume of the solid obtained by rotating the region bounded by $y = x^2$ and $y = 2x$ about the line $x = -1$.
The region bounded by $y = 4 - x^2$ and $y = 0$ is rotated about the line $x = 1$.
- Sketch the region and the axis. Verify that the region spans both sides of $x = 1$.
- Set up the volume integral using shells. (Hint: You'll need to split the integral.)
- Evaluate the integral to find the volume.
Common Mistakes
| Mistake | Why It Happens | Correction |
|---|---|---|
| Using $x - k$ when $x < k$ | Not checking which side of axis | Always sketch! If region is left of axis, use $k - x$. |
| Forgetting to change strip direction for horizontal axes | Habit from $y$-axis problems | For horizontal axes, use horizontal strips and integrate with $dy$. |
| Wrong integration variable | Confusion between cases | Vertical axis → $dx$; horizontal axis → $dy$. |
| Not splitting when region spans axis | Treating $\lvert x - k \rvert$ as $x - k$ | If region is on both sides of axis, split the integral at the axis. |
| Forgetting to convert bounds for $y$ | Using $x$-bounds when integrating $dy$ | Re-express bounds in terms of the new variable. |
Still Confused?
- Standard $y$-axis case unclear? → Review Shell Method: y-axis
- Distance/absolute value confusing? → Remember: distance is always positive. Sketch the axis and region!
- Not sure which strip direction? → Rule: strips are parallel to the axis for shells
- When to split integrals? → Only when the region spans across the axis
Common Misconceptions
for rotation about the horizontal axis $y = k$, vertical strips still create shells and the integration variable is $x$.
This is the iconic-graph error. For a horizontal axis of rotation, the strips that generate shells must be horizontal, so the integration variable is $y$ and the curves must be expressed as functions of $y$. For the region under $y = \sqrt{x}$ rotated about the $x$-axis, using vertical strips and integrating with $dx$ produces washers, not shells. To use shells for this rotation, horizontal strips at height $y$ are used with radius $y$, width $4 - y^2$, and integration over $y \in [0, 2]$: the strip direction determines the integration variable, not the equation form of the curve.
Mastery Checklist
Looking Ahead
You now have the tools to use shells with any axis. The final skill brings everything together:
- Shells vs. Washers: how to decide which method to use for any given problem
Mental Model
The Distance Rule:
No matter where the axis is, ask yourself: “How far is my shell from the axis?” That distance is the radius.
Think of it like measuring distance to a wall:
- It doesn’t matter which side you’re on
- Distance is always positive
- The formula changes based on which side ($x - k$ vs. $k - x$)
Draw the axis, draw your strip, and measure the gap.
| Previous | Up | Next |
|---|---|---|
| Shell Method: y-axis | Section 5.3 | Shells vs Washers |
Last updated: 2026-01-23