x=r cosθ
y=r sinθ (θ从0到π/4;r从1到2;积分区域是一个扇形)
则被积函数arctan(y/x)=arctan[(r sinθ/(r cosθ)]=arctan tanθ =θ
∫∫arctan(y/x)dσ
=∫(从0到π/4)dθ ∫(从1到2)r·θ dr
=∫(从0到π/4)[(1/2)r^2|(从1到2)]·θ dθ
=∫(从0到π/4)(3/2)θ dθ
=(3/2)·(1/2)θ^2|(从0到π/4)
=3π^2/64
x=r cosθ
y=r sinθ (θ从0到π/4;r从1到2;积分区域是一个扇形)
则被积函数arctan(y/x)=arctan[(r sinθ/(r cosθ)]=arctan tanθ =θ
∫∫arctan(y/x)dσ
=∫(从0到π/4)dθ ∫(从1到2)r·θ dr
=∫(从0到π/4)[(1/2)r^2|(从1到2)]·θ dθ
=∫(从0到π/4)(3/2)θ dθ
=(3/2)·(1/2)θ^2|(从0到π/4)
=3π^2/64