Find the exact length of the polar curve r = 2(1 + cos(θ)).

Answer 1

L = ∫₀^2π(r² + (r')²)^1/2dθ = ∫₀^2π(4 + 8cosθ + 4cos²θ + 4sin²θ)^1/2dθ = ∫₀^2π4Icosθ/2Idθ = 4∫₀^πcosθ/2dθ - 4∫_π^2πcosθ/2dθ = 8sinθ/2₀^π - 8sinθ/2_π^2π = 8 + 8 = 16