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Volume of Revolution Online Calculator

Volume-of-revolution calculator. Enter y=f(x) on an interval, rotate about the x-axis, compute V=π∫f²dx by disks.

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V = π ∫ₐᵇ [f(x)]² dx
Supports + - * / ^ ( ) and sin cos tan sec cot sqrt exp log abs asin atan pi e; write 2*x, not 2x
Lower limit of integration
Upper limit of integration

📖 Tutorial | Volume of Revolution

1. Definition
Rotating y=f(x) (f≥0) about the x-axis gives the solid volume by the disk method: V=π∫ₐᵇ [f(x)]²dx. Each slice is a disk of radius f(x) and area πf².
2. Symbols
SymbolsMeaning
f(x)Disk radius
π[f(x)]²Disk area
[a,b]Interval
VVolume
3. How it works
  • Builds the integrand [f(x)]²;
  • Multiplies by π and integrates adaptively;
  • Returns the volume.
4. Steps
  1. Enter y=f(x), e.g. x;
  2. Enter [a,b];
  3. Click Calculate for V.
5. Example
Example: y=x about x-axis on [0,1]: V=π/3≈1.047198 (cone = ⅓πr²h). Tool: 1.047198.
6. Pitfalls
The curve should stay on one side of the x-axis for the disk method;
This is a solid, not a shell;
The axis is fixed to the x-axis.

❓ FAQ | Volume of Revolution

Disk vs. shell method?
Disk slices perpendicular (πf²dx); shell uses thin cylinders parallel (2πx·f dx).
What solid is y=x rotated?
A cone; on [0,1] volume π/3.
Why f²?
Disk area is πr² with r=f(x).
What if the curve crosses below the axis?
Use |f| or washers; this tool squares f automatically.
Which functions?
+−*/^ and common elementary functions.