The area bounded by the 2 parabolas is A(θ) = 1/2∫(r₂²- r₁²).dθ between limits θ = a,b...
the limits are solution to 3cosθ = 1+cosθ the points of intersection of curves.
2cosθ = 1 => θ = ±π/3
A(θ) = 1/2∫(r₂²- r₁²).dθ = 1/2∫(3cosθ)² - (1+cosθ)².dθ
= 1/2∫(3cosθ)².dθ - 1/2∫(1+cosθ)².dθ
= 9/8[2θ + sin(2θ)] - 1/8[6θ + 8sinθ +sin(2θ)] ..
.............where I have used ∫(cosθ)².dθ=1/4[2θ + sin(2θ)]
= 3θ/2 +sin(2θ) - sin(θ)
Area = A(π/3) - A(-π/3)
= 3π/6 + sin(2π/3) -sin(π/3) - (-3π/6) - sin(-2π/3) + sin(-π/3)
= π.
