Volume of revolution of r = f(θ) about the initial line is

(2π/3) integral r^3 sin θ dθ

(2π/3) integral r^3 cos θ dθ

Volume of cardioid r=a(1+cosθ) about initial line

Short answer

Volume V = (2π/3) ∫₀^π r³ sinθ dθ. For r = a(1+cosθ), let u = 1+cosθ. Then V = (2πa³/3) ∫₀² u³ du = (2πa³/3)·4 = 8πa³/3.

Find the volume of the solid formed by the revolution of the cardioid r = a(1+cosθ) about the initial line.

For a curve r = f(θ) rotated about the initial line (θ = 0), the volume is

V = (2π/3) ∫ r³ sin θ dθ.

For the cardioid r = a(1 + cos θ), the upper half 0 ≤ θ ≤ π generates the whole solid.

V = (2π/3) ∫₀^π a³(1 + cosθ)³ sinθ dθ

Put u = 1 + cos θ, du = −sin θ dθ. Limits: θ = 0 gives u = 2; θ = π gives u = 0.

V = (2π a³/3) ∫₂⁰ u³ (−du)
  = (2π a³/3) ∫₀² u³ du
  = (2π a³/3) · [u⁴/4]₀²
  = (2π a³/3) · 4
  = 8π a³/3

Answer: V = 8πa³/3 cubic units.

✓ Answer checked by Claude Opus 5.5 (high effort)