What is the formula for calculating the required area?

因 cos(-2θ) = cos(2θ) = cos(2(π-θ)), 故極座標曲線 r = 2 + cos(2θ) 對稱於 θ=0 (x-軸) 也對稱於 θ=π/2 (y-軸). 所以只需 考慮 0 ≦ θ ≦ π/2, 再將結果乘以 4 即可. 又, r = 2 + cos(2θ) ≧ 1, 所以要計算面積的區域在單 位圓外, 上列曲線之內. 所以, 所求區域面積是 4 ∫_[0,π/2] [(2 + cos(2θ) )^2 - 1]/2 dθ = 2 ∫_[0,π/2] 3+4cos(2θ)+cos^2(2θ) dθ = ∫_[0,π/2] 7+8cos(2θ)+cos(4θ) dθ = [7θ + 4sin(2θ)+sin(4θ)/4]_0^{π/2} = 7π/2