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Calculus learning

Volumes of Revolution: Disk, Washer and Shell Setup

Volumes of revolution begin with a cross-section, not with an integral you already know. A disk has one radius; a washer removes an inner disk. The method and integration variable depend on the direction of the rotation axis.

Direct answer

About the x-axis with nonnegative radii, V=π∫ₐᵇ [R(x)²−r(x)²] dx.

Use perpendicular slices for disks/washers and parallel slices for shells. Distances are measured from the stated rotation axis.

Volumes of Revolution worked method

Rotate the region under y=x on [0,1] about the x-axis. Each perpendicular slice is a disk of radius x, so its area is πx².

Integrating the disk areas gives π[x³/3]₀¹=π/3 cubic units. The height x is squared before integration.

For outer radius 2 and inner radius x on [0,1], integrate π(4−x²), which gives 11π/3. Squaring the difference (2−x) would describe a different region.

A shell instead has circumference 2π times radius and height. Rotating under y=x around the y-axis on [0,1] gives 2π∫₀¹ x·x dx=2π/3.

For an axis y=k, radii are distances |f(x)−k|. The preview tool handles only the x-axis with ordered nonnegative polynomial radii; translate other setups yourself only when mathematically justified.

1
Vdisk=π∫₀¹ x² dx=π/3
2
Vwasher=π∫₀¹ (4−x²) dx=11π/3
3
Vshell=2π∫₀¹ x² dx=2π/3

Worked examples

Under y=x about x-axis, [0,1]
π3\frac{\pi}{3}

Disk radius is x.

Outer radius 2, inner radius x, [0,1]
11 π3\frac{11\,\pi}{3}

Subtract squared radii.

Under y=x about y-axis, [0,1]
2 π3\frac{2\,\pi}{3}

Use cylindrical shells, not the same disk integrand.

Common mistakes

  • • Using area instead of cross-section area.
  • • Squaring R−r rather than subtracting R²−r².
  • • Omitting the rotation axis.

Try the supported calculator preset

The preset opens a supported expression with its variable and conditions. Teaching examples elsewhere in this guide are reference explanations, not claims that the engine supports every method.

The preset is the disk cross-section πx², not the original curve x. The geometric setup must be chosen before integrating.

Open the area and volume application tool定积分计算器:π x2\pi\,x^{2}
Open the calculator without a preset: 定积分计算器

References

Frequently asked questions

When is the inner radius zero?+

When the region reaches the rotation axis without a hole in that slice.

Are shell and washer answers different?+

Correct setups for the same solid agree; different axes can describe different solids.

Can the current tool rotate about any axis?+

No. Its supported volume mode is x-axis disks/washers with nonnegative quadratic-or-lower radii.

Conclusion

Volumes of Revolution starts with the stated domain and conditions. Recheck those before carrying a worked example into a different problem.

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