# What is the washer 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 horizontal line y = k, by washers or disks.

- Page: https://www.acalculator.org/math/washer-method-calculator
- JSON spec: https://www.acalculator.org/math/washer-method-calculator.json
- Version: 7c499b47122f

## Default answer

Example with the default inputs (Outer curve y = f(x) x, Inner curve y = g(x) (empty is 0) 1/x, From x = a 1, To x = b 4, Axis y = k 0): The washer method volume for y = x from x = 1 to x = 4 about the line y = 0 is 63.61725124.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| f | Outer curve y = f(x) | One edge of the region. |
| g | Inner 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 y = k | Axis y = k: a number or a constant such as pi. |

## Outputs

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

## Method

V = π ∫ₐᵇ |(f(x) − k)² − (g(x) − k)²| dx: washers with outer and inner radii |f − k| and |g − k|.

## Assumptions

- The region lies on one side of the axis y = k.
- Angles in radians; an answer that fails its check is not shown.

## Worked examples

1. f = x, g = 1/x, a = 1, b = 4, k = 0 gives volume = 63.617251, exact = 81π/4. Source: OpenStax, Calculus Volume 1, section 6.2 Determining Volumes by Slicing (https://openstax.org/books/calculus-volume-1/pages/6-2-determining-volumes-by-slicing), Example 6.10 (washers about the x-axis; the integral set up there, π∫₁⁴ (x² − 1/x²) dx, has f(x) = x).
2. f = 4 - x, a = 0, b = 4, k = -2 gives volume = 167.551608, exact = 160π/3. Source: OpenStax, Calculus Volume 1, section 6.2 Determining Volumes by Slicing (https://openstax.org/books/calculus-volume-1/pages/6-2-determining-volumes-by-slicing), Example 6.11 (about the line y = −2).
3. f = sqrt(x), a = 1, b = 4, k = 0 gives volume = 23.561945, exact = 15π/2. Source: OpenStax, Calculus Volume 1, section 6.2 Determining Volumes by Slicing (https://openstax.org/books/calculus-volume-1/pages/6-2-determining-volumes-by-slicing), Example 6.8 (disks).

## FAQ

### What is the washer method?

A way to find the volume of a solid of revolution. Each thin vertical slice of the region turns into a washer: a disk of outer radius R with a hole of inner radius r. Its volume is about π(R² − r²) dx, and V = π ∫ₐᵇ (R(x)² − r(x)²) dx.

### What is the difference between the disk and washer methods?

The disk method is the washer method with no hole: the region touches the axis, so r = 0 and V = π ∫ R(x)² dx. Leave the inner curve empty (with the axis y = 0) for disks.

### How do I find R and r about a line y = k?

The radii are distances to the axis: R = |f(x) − k| and r = |g(x) − k|. About y = −2, the region under y = 4 − x on [0, 4] has R = 4 − x + 2 = 6 − x and r = 0 + 2 = 2.

### Which curve is the outer one?

It does not matter what you type first. The page uses |(f − k)² − (g − k)²|, so the curve farther from the axis is the outer radius at each x, even where the curves cross.

### Why does the page refuse a region that crosses the axis?

If the region lies on both sides of y = k, the slice has no hole: it is a full disk of the larger radius, and the washer formula would give the wrong volume. Split the region at the axis and use the side that sweeps farther.

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

Use the shell method calculator: it turns a region in x about a line x = k, with no need to solve the curves for x.

### 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 6.2 Determining Volumes by Slicing (retrieved 2026-10-03): https://openstax.org/books/calculus-volume-1/pages/6-2-determining-volumes-by-slicing