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Surface Area of Revolution Online Calculator

Surface-of-revolution area calculator. Enter y=f(x) on an interval, rotate about the x-axis, compute S=2π∫|f|√(1+f′²)dx.

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S = 2π ∫ₐᵇ |f(x)|√(1+[f′]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 | Surface Area of Revolution

1. Definition
Rotating y=f(x) (f≥0) about the x-axis gives surface area S=2π∫ₐᵇ f(x)√(1+[f′]²)dx, summing frustum strips.
2. Symbols
SymbolsMeaning
f(x)Radius of rotation
2πf(x)Circumference
√(1+f′²)dxGeneratrix element
SSurface area
3. How it works
  • Numerically finds f′ and builds |f|√(1+f′²);
  • Multiplies by 2π and integrates adaptively;
  • Returns the surface area.
4. Steps
  1. Enter y=f(x), e.g. x;
  2. Enter [a,b];
  3. Click Calculate for S.
5. Example
Example: y=x about x-axis on [0,1]: S=π√2≈4.442883. Tool: 4.442883.
6. Pitfalls
This is the lateral area, not the end caps;
|f| handles negative radii;
The axis is fixed to the x-axis.

❓ FAQ | Surface Area of Revolution

Surface area vs. volume?
Area is the shell (2πf·arc element); volume is the solid (πf²·thickness).
Why times 2π?
Each point traces a circle of circumference 2π·radius.
What does rotating y=x give?
A cone lateral surface: π√2 on [0,1].
Are end caps included?
No, only the generated lateral surface.
Which functions?
+−*/^ and common elementary functions.