求大师赐教！r=a(1+sinθ)所围图形的面积。

Question

如题。求帮助啊啊啊

Answer 1

面积=S_{θ:0->2PI}(1/2)r^2dθ=S_{θ:0->2PI}(1/2)a^2(1+sin(θ))^2dθ
=(1/2)a^2S_{θ:0->2PI}[1+2sin(θ) + (sin(θ))^2]dθ
=a^2/2S_{θ:0->2PI}[1+2sin(θ)]dθ + a^2/2S_{θ:0->2PI}[1-cos(2θ)]/2dθ
=a^2/2[θ - 2cos(θ)]_{θ:0->2PI} + a^2/4[θ - (1/2)sin(2θ)]_{θ:0->2PI}
=a^2/2[2PI-2+2] + a^2/4[2PI]
=(3PI/2)a^2

其中，PI=3.1415926