Arc Length Integral Calculator

How to Use This Calculator

1. Enter the function f(x) (e.g. x^2, sin(x)).
2. Set the bounds a and b.
3. Set subdivisions n (default 1000 gives high accuracy).
4. Get arc length L, chord length, and curve/chord ratio.
5. Use Parametric or Polar tabs for those curve types.

Formula

Example

Frequently Asked Questions

• What is the arc length formula for y=f(x)? ▼
  L = ∫ₐᵇ √(1 + (f'(x))²) dx. The integrand is the length of an infinitesimal tangent segment ds = √(dx²+dy²) = √(1+(dy/dx)²) dx.
• How do I calculate arc length of a parametric curve? ▼
  For x=x(t), y=y(t) from t=a to t=b: L = ∫ₐᵇ √((dx/dt)² + (dy/dt)²) dt. A circle of radius r traced once gives L = 2πr.
• What is the polar arc length formula? ▼
  For r=f(θ) from θ₁ to θ₂: L = ∫ √(r² + (dr/dθ)²) dθ. For a circle r=a, dr/dθ=0, so L = ∫a dθ = a(θ₂−θ₁) = 2πa for a full circle.
• What is the surface area of revolution? ▼
  Rotating y=f(x) about the x-axis generates surface area S = 2π ∫ₐᵇ |f(x)| √(1+(f'(x))²) dx. This uses the arc length element ds times the circumference 2π|y|.
• Why is arc length often hard to compute analytically? ▼
  The integrand √(1+(f')²) rarely simplifies to a closed form. Most arc length problems require numerical integration. Simple exceptions include lines (f'=constant) and some trigonometric/hyperbolic functions.

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