Volume about the initial line is

(2π/3) integral r^3 sin θ dθ

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

Short answer

Use V = (2π/3)∫ r³ sinθ dθ. With r = a(1-cosθ), substitute u = 1-cosθ to get (2πa³/3)∫₀² u³ du = 8πa³/3 cubic units.

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

For a curve r = f(θ) revolved about the initial line, V = (2π/3) ∫ r³ sin θ dθ.

Cardioid: r = a(1 − cos θ). The upper half (0 ≤ θ ≤ π) generates the full solid.

V = (2π/3) ∫₀^π a³(1 − cosθ)³ sinθ dθ

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

V = (2πa³/3) ∫₀² u³ du
  = (2πa³/3) · [u⁴/4]₀²
  = (2πa³/3) · 4

V = 8πa³/3 cubic units.

✓ Answer checked by Claude Opus 5.5 (high effort)