Surface area of cardioid by revolution

Reading: the surface generated when the cardioid r = a(1 + cos θ) is revolved about the initial line (θ = 0). The upper half 0 ≤ θ ≤ π generates the whole surface.

For revolution about the initial line, S = 2π ∫ y ds = 2π ∫ r sin θ · √(r² + (dr/dθ)²) dθ.

With r = a(1 + cos θ), dr/dθ = −a sin θ:

r² + (dr/dθ)² = a²[(1 + cos θ)² + sin²θ] = 2a²(1 + cos θ) = 4a² cos²(θ/2)

so ds = 2a cos(θ/2) dθ. Also r = 2a cos²(θ/2) and sin θ = 2 sin(θ/2) cos(θ/2).

S = 2π ∫₀^π 2a cos²(θ/2) · 2 sin(θ/2) cos(θ/2) · 2a cos(θ/2) dθ
  = 16π a² ∫₀^π cos⁴(θ/2) sin(θ/2) dθ

Put u = cos(θ/2), du = −½ sin(θ/2) dθ. Limits: θ = 0 → u = 1; θ = π → u = 0.

S = 16πa² · 2 ∫₀¹ u⁴ du
  = 32πa² · (1/5)

S = 32πa²/5 square units.