Volume of Revolution Tool

Math Calculus • Applications of Integrals

Written by STEM Calculators Team Published December 28, 2025 Updated February 24, 2026

2. Volume Of Revolution Tool

Computes volumes of solids of revolution using Disk, Washer, or Shell methods (including shifted axes), shades the revolved region on the 2D graph, and shows a true 3D rotating preview.

Inputs

ready

Method

Disk/Washer: \(V=\pi\int(R^2-r^2)\). Shell: \(V=2\pi\int(\text{radius})(\text{height})\).

Axis of revolution Axis shift \(c\)

Used only for shifted axes \(y=c\) or \(x=c\). Supports: pi, e, + − * / ^ and parentheses.

Slice / representation

Disk/Washer prefers slices perpendicular to the axis. Shell prefers slices parallel to the axis.

Output format Tolerance (numeric)

Adaptive Simpson tolerance (smaller = more accurate, slower).

Curve 1 (outer / top / right)

Use variable x for x-slices or y for y-slices. Implicit multiplication allowed: 2x, xsin(x), (x+1)(x-1).

Curve 2 (inner / bottom / left) or 0

Set to 0 if no inner curve is needed (e.g., disk method).

Lower bound \(a\)

Supports: pi, e, + − * / ^ and parentheses.

Upper bound \(b\)

Example: b = 4.

Tip: For y = √x on [0,4] about the x-axis, use Disk and x-slices: \(V=\pi\int_0^4(\sqrt{x})^2\,dx\).

Options

Show grid

Shade region (on graph)

Show curves

Show axis line

Show steps

Show solid preview (3D)

Presets

Click a preset to load, then Evaluate.

Graph

curve 1 curve 2 shaded region axis

Drag to pan • wheel/pinch to zoom. The shaded region is the 2D region being revolved on \([a,b]\) (or \([a,b]\) in the chosen slice variable).

x: —, y: — zoom(px/unit): —

Tip: If values blow up (near asymptotes), zoom out, pan away, or reduce the interval.

Solid preview (3D rotation)

mode: —

A true 3D mesh preview (illustrative, not to scale). Uses the current method and bounds.

angle

mesh: —

Result

—

Enter curves and bounds, then press Evaluate.

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2. Volume Of Revolution Tool — Theory

This tool computes the volume of a 3D solid obtained by revolving a planar region around an axis. It supports the Disk, Washer, and Shell methods, including shifted axes \(y=c\) or \(x=c\).

1) Disk & Washer methods

If cross-sections perpendicular to the axis are circles, the volume comes from adding up thin disks or washers.

\[ V = \pi \int_a^b \big(R(t)^2 - r(t)^2\big)\,dt \]

\(R(t)\) = outer radius (distance from the outer curve to the axis).
\(r(t)\) = inner radius (distance from the inner curve to the axis). For a disk (no hole), \(r(t)=0\).
Use x-slices when rotating around a horizontal axis (like \(y=0\) or \(y=c\)).
Use y-slices when rotating around a vertical axis (like \(x=0\) or \(x=c\)).

2) Shell method

If cross-sections parallel to the axis are cylindrical shells, each slice forms a thin shell with radius = distance to the axis and height = difference between curves.

\[ V = 2\pi \int_a^b \big(\text{radius}(t)\big)\big(\text{height}(t)\big)\,dt \]

About \(x=c\) using x-slices: radius \(=|x-c|\), height \(=|y_{\text{top}}-y_{\text{bot}}|\).
About \(y=c\) using y-slices: radius \(=|y-c|\), height \(=|x_{\text{right}}-x_{\text{left}}|\).

3) Shifted axes

Revolving around \(y=c\) or \(x=c\) is handled by measuring distances to that line:

\[ \text{distance to } y=c \text{ is } |y-c|,\qquad \text{distance to } x=c \text{ is } |x-c|. \]

4) Example (from the prompt)

Volume of \(y=\sqrt{x}\) from \(x=0\) to \(x=4\) revolved about the x-axis (Disk method):

\[ V = \pi\int_0^4 (\sqrt{x})^2\,dx = \pi\int_0^4 x\,dx = \pi\left[\frac{x^2}{2}\right]_0^4 = 8\pi. \]

5) What the tool shades and what the 3D preview means

The 2D graph shading shows the planar region between Curve 1 and Curve 2 over the chosen bounds.
The 3D preview is a mesh approximation of the revolved solid. It is illustrative (not a CAD model).

Troubleshooting tips

If you see Non-finite sample values, your function may be undefined on part of \([a,b]\) (e.g., \(\sqrt{x}\) for \(x<0\), division by zero, etc.).
If the graph looks “empty,” click Auto fit or zoom out and pan (the region might be off-screen).
Use the correct variable for your slice type: x-slices require expressions in x, y-slices require expressions in y.

Frequently Asked Questions

What is the disk method for volume of revolution?

The disk method applies when the cross-sections perpendicular to the axis are solid circles. The volume is V = pi * integral_a^b (R(t)^2) dt, where R(t) is the distance from the curve to the axis.

How is the washer method different from the disk method?

The washer method is used when the solid has a hole, so each cross-section is a washer instead of a full disk. The volume is V = pi * integral_a^b (R(t)^2 - r(t)^2) dt, where r(t) is the inner radius.

When should I use the shell method?

Use the shell method when slices parallel to the axis produce cylindrical shells, often avoiding complicated splitting. The volume is V = 2 * pi * integral_a^b (radius(t) * height(t)) dt.

How do shifted axes x = c or y = c change the setup?

Radii are measured as distances to the shifted line, using |x - c| or |y - c| depending on the slice variable and axis direction. The calculator uses the chosen c value to build the correct radius terms in the integral.

Why does changing the tolerance affect the computed volume?

The integral is evaluated numerically, so a smaller tolerance typically increases accuracy but may require more sampling and take longer. A larger tolerance can be faster but may reduce precision for sharp curvature or near-singular behavior.