Quick Reference $$\boxed{V = \int_a^b 2\pi x f(x) \, dx}$$
Component Meaning $x$ Radius (distance from strip to $y$-axis) $f(x)$ Height (length of vertical strip) $dx$ Thickness $2\pi x$ Circumference of the shell
In $V = 2\pi r h \Delta r$:
If this is unclear, review Shell Method Formula.
Solution: Set $x^2 = 2x$, so $x^2 - 2x = 0$, giving $x(x - 2) = 0$. Thus $x = 0$ or $x = 2$.
If this was difficult, you'll struggle with bounds for shell integrals. Review solving systems of equations.
Solution: $\left[\frac{x^{5/2}}{5/2}\right]_0^2 = \frac{2}{5} \cdot 2^{5/2} = \frac{2}{5} \cdot 4\sqrt{2} = \frac{8\sqrt{2}}{5}$
If fractional exponents feel unfamiliar, review Power Rule Integration.
Rotation about the $y$-axis is the natural habitat for the shell method. When a region defined by $y = f(x)$ rotates about the $y$-axis, vertical strips become cylindrical shells in a direct way:
This is often simpler than the disk/washer method, which would require solving $y = f(x)$ for $x$—impossible or painful when $f$ is a cubic or worse.
| Property | Value |
|---|---|
| Chapter | 5 - Applications of Integration |
| Section | 5.3 |
| Difficulty | Intermediate |
| Time | ~20 minutes |
| Scenario | Shell Formula |
|---|---|
| Region under $y = f(x)$ | $V = \int_a^b 2\pi x f(x) \, dx$ |
| Region between $y = f(x)$ and $y = g(x)$ | $V = \int_a^b 2\pi x [f(x) - g(x)] \, dx$ |
For a region bounded by $y = f(x)$ (where $f(x) \geq 0$), $y = 0$, $x = a$, and $x = b$ (with $0 \leq a < b$), rotated about the y-axis:
$$\boxed{V = \int_a^b 2\pi x f(x) \, dx}$$
| Component | What It Represents | In the Formula |
|---|---|---|
| Radius | Distance from $y$-axis to strip | $x$ |
| Height | Length of vertical strip | $f(x)$ |
| Thickness | Width of strip | $dx$ |
| Circumference | Perimeter of circular cross-section | $2\pi x$ |
Step 1: Sketch the region and identify the bounds on $x$.
Step 2: Draw a typical vertical strip at position $x$.
Step 3: Identify the shell components:
Step 4: Write the integral: $$V = \int_a^b 2\pi \cdot (\text{radius}) \cdot (\text{height}) \, dx$$
Step 5: Simplify the integrand before integrating.
Step 6: Evaluate the definite integral.
Step 7: Verify (optional but recommended): Check dimensions or compare with another method.
y
│ ╭───────╮
│ ╱ │
│ ╱ f(x) │ ← height of shell = f(x)
│ ╱ │
│ ╱───────────────┤
├─────┬───────────┼─── x
0 a x b
↑
└── radius of shell = x (distance to y-axis)
When this strip rotates around the y-axis,
it sweeps out a cylindrical shell.When the region is between $y = g(x)$ (bottom) and $y = f(x)$ (top):
$$V = \int_a^b 2\pi x [f(x) - g(x)] \, dx$$
The height of each shell is the vertical distance between the curves.
y
│ ╭─── f(x) (top curve)
│ ╱
│ ╱ ↕ height = f(x) - g(x)
│ ╱
│ ╲─────── g(x) (bottom curve)
├─────────────────── x
0 a x bThe shell method was developed because many natural shapes—like solids formed by rotating cubics such as $y = x^3 - x^2 + x$—lead to equations that are impractical to solve when using the disk method. Before computer algebra systems, mathematicians needed methods that worked with the "natural" description of a region. The shell method lets you work with $y = f(x)$ directly, without solving for $x = f^{-1}(y)$.
Find the volume of the solid obtained by rotating the region under $y = x^2$ from $x = 0$ to $x = 3$ about the $y$-axis.
Find the volume of the solid obtained by rotating the region bounded by $y = 3x - x^3$ and $y = 0$ (for $x \geq 0$) about the $y$-axis.
Find the volume of the solid obtained by rotating about the $y$-axis the region bounded by $y = 2x$ and $y = x^2$.
Find the volume of the solid obtained by rotating about the $y$-axis the region bounded by $y = 3x^2 - x^3$ and $y = 0$.
Consider the region bounded by $y = \sqrt{x}$, $y = 0$, and $x = 4$, rotated about the $y$-axis.
| Mistake | Why It Happens | Correction |
|---|---|---|
| Using $2\pi f(x)$ as radius | Confusing radius with height | Radius is $x$ (distance to axis), height is $f(x)$. |
| Forgetting the $2\pi$ | Rushed setup or mixing with disk method | Always write out: circumference × height × thickness = $2\pi x \cdot f(x) \cdot dx$. |
| Wrong bounds (using $y$-values) | Mixing up integration variables | When integrating with $dx$, bounds are $x$-values. |
| Height as $(f(x))^2$ | Confusing with the disk method | The disk method squares the radius; the shell method doesn't square the height. |
| Subtracting curves in wrong order | Not checking which is "on top" | Test a point between intersections to see which $y$-value is larger. |
You've mastered the standard case. Next, we generalize:
The Vertical Strip Wrap:
Picture taking a tall, thin strip of paper and wrapping it around a pole (the $y$-axis). The distance from the strip to the pole is the radius, the strip's length is the height, and its width is the thickness. Sum up infinitely many such wrapped strips to get the solid.
When $y = f(x)$ is complicated, this "wrap around the axis" approach beats having to unwrap and re-describe the curves in terms of $y$.
| Previous | Up | Next |
|---|---|---|
| Shell Method Formula | Section 5.3 | Shell Method: Other Axes |
Last updated: 2026-01-23