Use the Parametric Equations of an Ellipse

Question

Use the parametric equations of an ellipse

x = 12cos⁡(theta)

y = 18sin⁡(theta)

0 ≤ (theta) ≤ 2pi



to find the area that it encloses.

Answer 1

Area A = 1/2∫₀^2π(xdy/dθ - ydx/dθ)dθ = 1/2∫₀^2π(12cosθ·18cosθ + 18sinθ·12sinθ)dθ = 108·2π = 216π

Answer 2

cosθ = x/12 and sinθ = y/18

Since cos²θ + sin²θ = 1, we have x² / 12² + y² / 18² = 1

a = 18 and b = 12

Area = πab = 216π