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Washer method calculator

Enter the curves that bound a region, the interval and the axis. You get the outer and inner radius, the integral set up, the volume (exact in π when it's a simple multiple) and a sketch of the region.

By the Calcist team · Checked against known answers · Updated October 29, 2026

About the x-axis or a horizontal line, curves are y = f(x). About the y-axis or a vertical line, write them as x = f(y) and use y. Which curve is outer is worked out for you.

Private: calculated on your device. Nothing you type is sent anywhere.

Next: Check the integral with the integral calculator, or review how definite integrals are evaluated in the fundamental theorem of calculus.

How the washer method works

Rotate a flat region around a line and it sweeps out a solid. Slice the solid perpendicular to the axis: each slice is a washer, a disk with a hole. Its area is π times the outer radius squared minus π times the inner radius squared. Adding up the slices is an integral.

V = π ∫ₐᵇ ( [R(x)]² − [r(x)]² ) dx

R(x) is the distance from the axis to the farther curve and r(x) is the distance to the nearer one. When the region touches the axis, r = 0 and this is the disk method, V = π∫R² dx. Rotating about a horizontal line y = k, the radii become |f(x) − k| and |g(x) − k|. About the y-axis or a vertical line, slice along y instead and write the curves as x = f(y).

Worked example

Rotate the region between y = x and y = x² for 0 ≤ x ≤ 1 about the x-axis.

  1. On [0, 1], x ≥ x², so the outer radius is R = x and the inner radius is r = x².
  2. V = π ∫₀¹ (x² − x⁴) dx.
  3. The antiderivative is x³/3 − x⁵/5, which is 1/3 − 1/5 = 2/15 at x = 1 and 0 at x = 0.
  4. V = 2π/15 ≈ 0.418879 cubic units.

Rotate the same region about y = −1 instead and both radii grow by 1: R = x + 1, r = x² + 1. Then V = π∫₀¹ ((x + 1)² − (x² + 1)²) dx = 7π/15 ≈ 1.466077. About the y-axis, the region between x = 2 and x = √y for 0 ≤ y ≤ 4 gives V = π∫₀⁴ (4 − y) dy = 8π ≈ 25.132741.

How it's calculated here

The calculator checks 201 points to decide which curve is farther from the axis, then integrates π(R² − r²) with adaptive Simpson's rule, reporting an error estimate. If the result is a whole-number fraction times π, it shows that exact form too. If the curves cross, it uses the farther one at every point; if the axis passes through the region, slices there are full disks.

Limitations

  • The region is the area between the two curves over your interval. To use the points where the curves meet as limits, find them first (for example with the equation solver).
  • About a vertical axis, curves must be written as x = f(y). If you only have y = f(x), the shell method, V = 2π∫ (radius)(height) dx, may be easier by hand.
  • Curves must be defined on the whole interval. Improper volumes (infinite bounds) aren't supported.

Need just the integral? Use the integral calculator, or see the area under each curve with the Riemann sum calculator.

Sources

How we checked this

Before every update, this calculator is run in a real browser with these known answers, and it must match all of them:

  • Between y = x and y = x² on [0, 1] about the x-axis: 2π/15 ✓
  • Disk: y = √x on [0, 4] about the x-axis: 8π ✓
  • Same region about y = −1: 7π/15 ✓
  • About the y-axis: x = 2 and x = √y on [0, 4]: 8π ✓
  • About y = 2: region under y = x² on [0, 1]: 17π/15 ✓
  • Edge case: equal bounds are rejected ✓

Last checked October 26, 2026. Spotted something wrong? The math is done in your browser with the formula shown on this page, so you can check any result by hand.

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Questions people ask

What is the washer method formula?

V = π∫(R² − r²) dx, where R is the outer radius and r the inner radius, both measured from the axis of rotation.

What's 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∫R² dx.

How do you use the washer method about the y-axis?

Slice horizontally. Write each curve as x = f(y), use y-limits, and integrate π(R(y)² − r(y)²) dy.

How do you rotate about a line like y = −1?

Measure radii from that line. For curves above y = −1, R = f(x) + 1 and r = g(x) + 1.

When should I use shells instead of washers?

When the curves are easy to write as y = f(x) but the axis is vertical (or the reverse). Shells use V = 2π∫(radius)(height) dx and avoid solving for x.

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