计算二重积分∫∫√xdxdy其中D={(x,y)|x^2+y^2<=x}

Question

如题所述

Answer 1

简单计算一下即可，答案如图所示

Answer 2

解：原式=∫<-π/2,π/2>dθ∫<0,cosθ>√(rcosθ)*rdr (作极坐标变换)
=∫<-π/2,π/2>√cosθdθ∫<0,cosθ>r^(3/2)dr
=(2/5)∫<-π/2,π/2>√cosθ*(cosθ)^(5/2)dθ
=(2/5)∫<-π/2,π/2>(cosθ)^3dθ
=(2/5)∫<-π/2,π/2>(1-(sinθ)^2)cosθdθ
=(2/5)∫<-π/2,π/2>(1-(sinθ)^2)d(sinθ)
=(2/5)(2-2/3)
=8/15。本回答被提问者采纳