Question

急急急！用定积分求极坐标图形面积（）

lz我是自学微积分，所以问题很弱智，求高手们稍微详细点的解答！谢谢了！
比如求r＝1+sin(Θ) r＝3sin（Θ）的重叠面积。
图形画出来就是求(0,Л/6)上r＝3sin（Θ）的面积加上 (Л/6,Л/2)上r＝1+sin(Θ)的面积。那是直接对r＝1+sin(Θ) r＝3sin（Θ）分别积分吗？

Answer 1

极坐标的面积微元是dA=(1/2)r²dθ
因此本题就是计算：
A=(1/2)∫[0--->π/6] 9sin²θ dθ+(1/2)∫[π/6--->π/2] (1+sinθ)² dθ
=(9/4)∫[0--->π/6] (1-cos2θ) dθ+(1/2)∫[π/6--->π/2] (1+2sinθ+sin²θ) dθ
=(9/4)(θ-(1/2)sin2θ) |[0--->π/6] + (1/2)∫[π/6--->π/2] (1+2sinθ+(1/2)(1-cos2θ)) dθ
=(9/4)(π/6-(1/2)(√3/2))+(1/2)∫[π/6--->π/2] ((3/2)+2sinθ-(1/2)cos2θ) dθ
=3π/8-9√3/16+(1/2)((3/2)θ-2cosθ-(1/4)sin2θ) |[π/6--->π/2]
=3π/8-9√3/16+(1/2)((3/2)(π/2)-0-0-(3/2)(π/6)+√3+(1/4)(√3/2))
=5π/8

Answer 2

建议你去高等数学吧

追问

世界上有种东西叫预习谢谢