Question

高等数学 定积分问题，求解

Answer 1

两条曲线围成的图形如下所示：

令 3cosθ=1+cosθ，解得两哗耐曲线交点的极角 θ=±π/3

两条曲线围成的图形关于极轴对称，其公共部分的面积 S 等于极轴乱没春上方公共部分面积S1+S2的二倍。

S1 = ∫ [0..π/3] 1/2 (1+cosθ)² dθ

= 1/2 ∫[0..π/3] [1+2cosθ+(1+cos2θ)/2] dθ

= 1/2 (3θ/2 + 2sinθ + 1/4 sin2θ) | [0..π/3] 

= π/4 + 9/8 sinπ/3

S2 = ∫ [π/3..π/2] 1/2 (3cosθ)² dθ }

=  察旅∫ [π/3..π/2] 9/2 (1+cos2θ)/2 dθ

= (9θ/4 + 9/8 sin2θ) | [π/3..π/2]

= 9/4 (π/2-π/3) + 9/8 (sinπ-sin2π/3)

= 3π/8 - 9/8 sin(π/3)

S = 2(S1+S2) = 2 [π/4 + 9/8 sinπ/3 + 3π/8 - 9/8 sin(π/3)] = 5π/4

Answer 2