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Area of one loop of r = a sin 3θ

Short answer

One loop spans θ=0 to π/3. A = ½∫₀^{π/3} a² sin²3θ dθ = πa²/12 square units.

Find the area of one loop of the curve r = a sin 3θ.

Curve: r = a sin 3θ, a three-petal rose. r = 0 when 3θ = 0, π, 2π, ..., so θ = 0 and θ = π/3 bound one loop.

Area = ½ ∫ r² dθ over the loop:

A = ½ ∫₀^(π/3) a² sin²3θ dθ
  = (a²/4) ∫₀^(π/3) (1 − cos 6θ) dθ
  = (a²/4) [θ − sin6θ/6]₀^(π/3)
  = (a²/4)(π/3 − 0)

Area of one loop = πa²/12 square units.

✓ Answer checked by Claude Opus 5.5 (high effort)

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