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Arc Length Online Calculator

Arc length calculator for plane curves. Enter y=f(x) and an interval to numerically compute L=∫√(1+f′²)dx, with a comparison value—great for integral-application homework.

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L = ∫ₐᵇ √(1 + [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 | Arc Length

1. Definition
The arc length of a smooth curve y=f(x) on [a,b] is L=∫ₐᵇ √(1+[f′(x)]²)dx, summing small segments √(Δx²+Δy²).
2. Symbols
SymbolsMeaning
f(x)Curve
f′(x)Slope of tangent
√(1+f′²)Arc element ds
[a,b]Interval
3. How it works
  • Computes f′(x) by central difference;
  • Builds ds=√(1+f′²) and integrates adaptively;
  • Returns the arc length.
4. Steps
  1. Enter y=f(x), e.g. x;
  2. Enter [a,b];
  3. Click Calculate for L.
5. Example
Example: y=x on [0,1]: f′=1, L=√2≈1.414214. Tool: 1.414214.
6. Pitfalls
The curve must be smooth and differentiable;
Arc length is path length, not straight-line distance;
f′ is a difference quotient; avoid undefined sample points.

❓ FAQ | Arc Length

How to remember the formula?
View ds as the hypotenuse: ds=√(dx²+dy²)=√(1+y′²)dx.
How does it find y′?
Central difference (f(x+h)−f(x−h))/(2h), h≈1e-6.
Arc length of y=x on [0,1]?
√2≈1.414.
Parametric curves?
This tool supports y=f(x) only.
Which functions?
+−*/^ and common elementary functions.