What is the washer method volume?
Type the outer and inner curves of a region, the interval [a, b], and a horizontal axis y = k. The page turns the region about the axis and gives the volume by washers (or disks, with no inner curve), exactly when it can.
- Volume
- 63.61725124
The washer method volume for y = x from x = 1 to x = 4 about the line y = 0 is 63.61725124.
- Exact volume
- 81π/4
Volume: 63.61725124. The washer method volume for y = x from x = 1 to x = 4 about the line y = 0 is 63.61725124.
How to calculate
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.
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.
Method: V = π ∫ₐᵇ |(f(x) − k)² − (g(x) − k)²| dx: washers with outer and inner radii |f − k| and |g − k|.
- 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
Each example is checked against the calculator on every build.
- 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 gives Volume 63.617251, Exact volume 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)
- Outer curve y = f(x) 4 - x, From x = a 0, To x = b 4, Axis y = k -2 gives Volume 167.551608, Exact volume 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)
- Outer curve y = f(x) sqrt(x), From x = a 1, To x = b 4, Axis y = k 0 gives Volume 23.561945, Exact volume 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)
How it works
The region lies between y = f(x) and y = g(x) for a ≤ x ≤ b (an empty g is the x-axis, g = 0). It turns about the horizontal line y = k. a, b and k are numbers or constants such as pi, with a < b.
V = π ∫ₐᵇ |(f(x) − k)² − (g(x) − k)²| dx.
- The region must lie on one side of the axis: the page finds where (f(x) − k)(g(x) − k) changes sign inside (a, b) (a grid of 8,001 points and bisection), and the product must not be negative at the middle of any piece between those points. Otherwise there is no answer. (With an empty g and k = 0 the product is 0, so disks below the axis are allowed: their radius is |f|.)
- (f − k)² − (g − k)² = (f − g)(f + g − 2k). The page finds where either factor changes sign inside (a, b) (where the curves swap which is farther from the axis), each point exact where the algebra solves it and the solution checks, on the same grid; two points within 10⁻¹² count once.
- Between each two such points, a computer algebra system (nerdamer, open source) finds ∫ (f(x) − g(x))(f(x) + g(x) − 2k) dx, checked against a numeric integral. Typed decimals go to the algebra as exact fractions (1e-9 as 1/10^9). The absolute values are added and multiplied by π.
- The exact volume is π times the sum of the exact pieces, each with the sign that makes it positive, simplified by the algebra. It shows when it holds no rounded number and equals the decimal volume to 10⁻⁹; its value is then the one shown.
Angles are in radians. There is no answer when f or g has a pole or is not real on [a, b], an end point included.
Worked examples by hand
Between y = x and y = 1/x on [1, 4], about the x-axis (OpenStax Calculus Volume 1, Example 6.10, whose set-up is π∫₁⁴ (x² − (1/x)²) dx). V = π [x³/3 + 1/x]₁⁴ = π (64/3 + 1/4 − 1/3 − 1) = 81π/4 ≈ 63.61725124.
y = 4 − x and the x-axis on [0, 4], about y = −2 (Example 6.11). R = 6 − x and r = 2. V = π ∫₀⁴ ((6 − x)² − 4) dx = π [−(6 − x)³/3 − 4x]₀⁴ = π (−8/3 − 16 + 72) = 160π/3 ≈ 167.5516082.
Disks: y = √x on [1, 4], about the x-axis (Example 6.8). V = π ∫₁⁴ x dx = π (16 − 1)/2 = 15π/2 ≈ 23.5619449.
Other questions people ask
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.