Question

选用适当的坐标计算下列各题：二重积分y^2/x^2dxdy,其中D是由曲线x^2+Y^2=2x所围成的平面区域。求过程，谢

Answer 1

解：原式=∫<-π/2,π/2>dθ∫<0,2cosθ>[(rsinθ)²/(rcosθ)²]rdr (作极坐标变换)
=∫<-π/2,π/2>(sin²θ/cos²θ)dθ∫<0,2cosθ>rdr
=2∫<-π/2,π/2>(sin²θ/cos²θ)cos²θdθ
=2∫<-π/2,π/2>sin²θdθ
=∫<-π/2,π/2>[1-cos(2θ)]dθ (应用倍角公式)
=π/2+π/2
=π。