CalcRef

Parametric, Polar & Vectors · Parametric Curves

Parametric Area

A=∫αβy(t) x′(t) dtA = \int_\alpha^\beta y(t)\, x'(t)\, dt

The area under a parametric curve. More precisely, the signed area between the curve and the x-axis.

Worked examples

Find the area enclosed by x = cos t, y = sin t (circle of radius 1).
  1. A = -∫₀^(2π) sin t · (-sin t) dt = ∫₀^(2π) sin²t dt
  2. = ∫₀^(2π) (1-cos2t)/2 dt = [t/2 - sin2t/4]₀^(2π) = π

Answer: π

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