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Parametric, Polar & Vectors · Polar Coordinates

Polar Arc Length

L=∫αβr2+(drdθ)2 dθL = \int_\alpha^\beta \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2}\, d\theta

The arc length of a polar curve r = f(θ) from θ = α to θ = β.

  • 2 worked examples
  • 12 in Parametric, Polar & Vectors

Worked examples

Find the circumference of r = 2 (circle).
  1. dr/dθ = 0. L = ∫₀^(2π) √(4+0) dθ = ∫₀^(2π) 2 dθ = 4π

Answer: 4π

1 more worked examplePremium

Find the length of the cardioid r = 1 + cos θ, 0 ≤ θ ≤ 2π.

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Common questions

What is the Polar Arc Length?

The arc length of a polar curve r = f(θ) from θ = α to θ = β. It is one of the parametric, polar & vectors formulas in the CalcRef reference.

How do you use the Polar Arc Length?

Worked example. Find the circumference of r = 2 (circle). dr/dθ = 0. L = ∫₀^(2π) √(4+0) dθ = ∫₀^(2π) 2 dθ = 4π. Answer: 4π.

Practice

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