CalcRef

Parametric, Polar & Vectors · Parametric Curves

Parametric Arc Length

L=∫αβ(dxdt)2+(dydt)2 dtL = \int_\alpha^\beta \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\, dt

The length of a parametric curve from t = α to t = β.

Variables

SymbolNameUnit
alphaStart parameter—
betaEnd parameter—

Worked examples

Find the circumference of x = cos t, y = sin t, 0 ≤ t ≤ 2π.
  1. dx/dt = -sin t, dy/dt = cos t
  2. √(sin²t + cos²t) = 1
  3. L = ∫₀^(2π) 1 dt = 2π

Answer: 2π

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