# What is the shell method volume?

Finds the volume of the solid made by turning the region between y = f(x) and y = g(x), a ≤ x ≤ b, about a vertical line x = k.

- Page: https://www.acalculator.org/math/shell-method-calculator
- JSON spec: https://www.acalculator.org/math/shell-method-calculator.json
- Version: fe15952c36dd

## Default answer

Example with the default inputs (Top curve y = f(x) 2x - x^2, From x = a 0, To x = b 2, Axis x = k 0): The shell method volume for y = 2x - x^2 from x = 0 to x = 2 about the line x = 0 is 8.37758041.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| f | Top curve y = f(x) | One edge of the region. |
| g | Bottom curve y = g(x) (empty is 0) | The other edge; empty means the x-axis. |
| a | From x = a | From x = a: a number or a constant such as pi. |
| b | To x = b | To x = b: a number or a constant such as pi. |
| k | Axis x = k | Axis x = k: a number or a constant such as pi. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| volume | Volume | 2π ∫ \|x − k\| \|f(x) − g(x)\| dx from a to b. |
| exact | Exact volume | The volume, exactly. |

## Method

V = 2π ∫ₐᵇ |x − k| |f(x) − g(x)| dx: shells of radius |x − k| and height |f − g|.

## Assumptions

- The axis x = k is outside the open interval (a, b).
- Angles in radians; an answer that fails its check is not shown.

## Worked examples

1. f = 2x - x^2, a = 0, b = 2, k = 0 gives volume = 8.37758, exact = 8π/3. Source: OpenStax, Calculus Volume 1, section 6.3 Volumes of Revolution: Cylindrical Shells (https://openstax.org/books/calculus-volume-1/pages/6-3-volumes-of-revolution-cylindrical-shells), Example 6.13 (about the y-axis).
2. f = x, a = 1, b = 2, k = -1 gives volume = 24.085544, exact = 23π/3. Source: OpenStax, Calculus Volume 1, section 6.3 Volumes of Revolution: Cylindrical Shells (https://openstax.org/books/calculus-volume-1/pages/6-3-volumes-of-revolution-cylindrical-shells), Example 6.15 (about the line x = −1).
3. f = sqrt(x), g = 1/x, a = 1, b = 4, k = 0 gives volume = 59.061942, exact = 94π/5. Source: OpenStax, Calculus Volume 1, section 6.3 Volumes of Revolution: Cylindrical Shells (https://openstax.org/books/calculus-volume-1/pages/6-3-volumes-of-revolution-cylindrical-shells), Example 6.16 (between √x and 1/x, about the y-axis).
4. f = 1/x, a = 1, b = 3, k = 0 gives volume = 12.566371, exact = 4π. Source: OpenStax, Calculus Volume 1, section 6.3 Volumes of Revolution: Cylindrical Shells (https://openstax.org/books/calculus-volume-1/pages/6-3-volumes-of-revolution-cylindrical-shells), Example 6.12.

## FAQ

### What is the shell method?

A way to find the volume of a solid of revolution. Cut the region into thin vertical strips; turning a strip at x about the axis makes a thin cylindrical shell of radius |x − k|, height |f(x) − g(x)| and thickness dx, with volume about 2π × radius × height × dx. Adding the shells gives V = 2π ∫ₐᵇ |x − k| |f(x) − g(x)| dx.

### When should I use the shell method instead of washers?

Use shells when the region is easy to describe with functions of x and the axis is vertical (the y-axis or x = k). Washers about a vertical axis would need the curves written as functions of y. For y = 2x − x² about the y-axis, shells need no inverse function.

### How do I revolve about the y-axis?

Use k = 0. The radius of each shell is then |x|. For f(x) = 2x − x² on [0, 2] that gives V = 2π ∫₀² x(2x − x²) dx = 8π/3.

### How do I revolve about another vertical line?

Type its x value as k. About x = −1, the radius is x + 1: the region under y = x on [1, 2] gives V = 2π ∫₁² (x + 1) x dx = 23π/3.

### Why must the axis not cut the region?

If the axis x = k lies strictly between a and b, the two sides of the region sweep through the same space, and adding their shells counts that space twice. Split the region at the axis and use the larger side, or choose a k outside (a, b).

### How do I revolve about a horizontal axis?

With shells, a horizontal axis needs the region written as functions of y. Use the washer method calculator instead: it turns a region in x about a line y = k.

### How is the answer checked?

Each definite integral comes from a computer algebra system and is checked against a numeric integral. The exact volume shows only when it equals the decimal volume to 10⁻⁹.

## Sources

- OpenStax, Calculus Volume 1, section 6.3 Volumes of Revolution: Cylindrical Shells (retrieved 2026-10-03): https://openstax.org/books/calculus-volume-1/pages/6-3-volumes-of-revolution-cylindrical-shells