Area common to circle and cardioid

Curves: circle r = a and cardioid r = a(1 + cos θ). They meet when 1 + cos θ = 1, so cos θ = 0, θ = ±π/2.

By symmetry about the initial line, the common area is twice the area for 0 ≤ θ ≤ π.

• For 0 ≤ θ ≤ π/2: cardioid radius ≥ a, so the circle is the inner boundary (r = a).
• For π/2 ≤ θ ≤ π: cardioid radius ≤ a, so the cardioid is the inner boundary.

Common area = 2 [ ½∫₀^(π/2) a² dθ + ½∫_(π/2)^π a²(1+cosθ)² dθ ]
            = a²·(π/2) + a² ∫_(π/2)^π (1 + 2cosθ + cos²θ) dθ

Evaluate: ∫(π/2)^π 1 dθ = π/2; ∫(π/2)^π 2 cos θ dθ = 2(0 − 1) = −2; ∫_(π/2)^π cos²θ dθ = π/4.

Sum = π/2 − 2 + π/4 = 3π/4 − 2.

Area = a² [π/2 + 3π/4 − 2] = a² (5π/4 − 2)

Common area = a²(5π − 8)/4 square units.