CalcRef

Parametric, Polar & Vectors · Polar Coordinates

Polar Area

A=12∫αβ[r(θ)]2 dθA = \frac{1}{2}\int_\alpha^\beta [r(\theta)]^2 \, d\theta

The area enclosed by a polar curve r = f(θ) from θ = α to θ = β.

Variables

SymbolNameUnit
alphaStart anglerad
betaEnd anglerad

Worked examples

Find the area enclosed by r = 2 (circle of radius 2).
  1. A = (1/2)∫₀^(2π) 4 dθ = 2(2π) = 4π

Answer: 4π

Find the area of one petal of r = sin(2θ).
  1. One petal: 0 ≤ θ ≤ π/2
  2. A = (1/2)∫₀^(π/2) sin²(2θ) dθ = (1/2)∫₀^(π/2) (1-cos4θ)/2 dθ
  3. = (1/4)[θ - sin4θ/4]₀^(π/2) = (1/4)(π/2) = π/8

Answer: π/8

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