Volume of Revolution Calculator

Name: Volume of Revolution Calculator
Availability: InStock
Rating: 4.6 (559 reviews)
Author: Generatr

Disk, washer & shell methods

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4.6(559 reviews)

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Revolution

V = π∫[f(x)]²dx

Function

Slices

Numerical

0.628266

Exact

0.628319

Error

5.2359e-5

Slices

x=0.005f=0.0000dV=0.000000

x=0.015f=0.0002dV=0.000000

x=0.025f=0.0006dV=0.000000

x=0.035f=0.0012dV=0.000000

x=0.045f=0.0020dV=0.000000

x=0.055f=0.0030dV=0.000000

x=0.065f=0.0042dV=0.000001

x=0.075f=0.0056dV=0.000001

x=0.085f=0.0072dV=0.000002

x=0.095f=0.0090dV=0.000003

x=0.105f=0.0110dV=0.000004

x=0.115f=0.0132dV=0.000005

x=0.125f=0.0156dV=0.000008

x=0.135f=0.0182dV=0.000010

x=0.145f=0.0210dV=0.000014

x=0.155f=0.0240dV=0.000018

x=0.165f=0.0272dV=0.000023

x=0.175f=0.0306dV=0.000029

x=0.185f=0.0342dV=0.000037

x=0.195f=0.0380dV=0.000045

x=0.205f=0.0420dV=0.000055

x=0.215f=0.0462dV=0.000067

x=0.225f=0.0506dV=0.000081

x=0.235f=0.0552dV=0.000096

x=0.245f=0.0600dV=0.000113

x=0.255f=0.0650dV=0.000133

x=0.265f=0.0702dV=0.000155

x=0.275f=0.0756dV=0.000180

x=0.285f=0.0812dV=0.000207

x=0.295f=0.0870dV=0.000238

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About Volume of Revolution Calculator

A volume of revolution calculator using disk, washer, and shell methods. Select from preset functions, set bounds, and choose axis of revolution. Shows the integral setup, Δx slices, and total volume. All calculations are client-side with numerical integration. Essential for calculus II and engineering.

Volume of Revolution Calculator Features

Disk method
Washer method
Shell method
Preset funcs
Step-by-step

Volume of revolution: rotate f(x) around an axis. Disk method: V = π∫[f(x)]²dx (solid). Washer method: V = π∫([R(x)]²−[r(x)]²)dx (hollow). Shell method: V = 2π∫x·f(x)dx (cylindrical shells). Choice depends on axis and function.

How to Use

Set up the problem:

Function: f(x) to rotate
Bounds: [a, b]
Method: Disk, washer, or shell

When to Use Each

Disk: solid, perpendicular slices
Washer: hollow, perpendicular slices
Shell: parallel to axis of revolution

Classic Examples

Sphere: rotate semicircle → V=4πr³/3. Cone: rotate line y=rx/h → V=πr²h/3. Torus: rotate circle off-axis → V=2π²Rr².

Step-by-Step Instructions

1Select a function.
2Set integration bounds.
3Choose method.
4View integral setup.
5Get volume.

Volume of Revolution Calculator — Frequently Asked Questions

When should I use shell vs. disk method?+

Use disk/washer when slicing perpendicular to the axis of revolution. Use shell when slicing parallel. If revolving around x-axis with y=f(x), disk is natural. Around y-axis with y=f(x), shell is easier.

Can I revolve around axes other than x or y?+

Yes. For y=a, replace f(x) with |f(x)−a| in the formulas. For x=b, adjust the shell radius to |x−b|. The principle is the same; only the distances change.

How is a torus formed?+

Rotate a circle of radius r centered at distance R from the axis (R > r). Volume = 2π²Rr² (Pappus' theorem: V = 2πR̄ × Area, where R̄ is the centroid distance).

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