用极坐标,x²+y²=2y的极坐标方程为：r=2sinθ
∫∫ xy dxdy
=∫∫ r³cosθsinθ drdθ
=∫[π/4→π/2] cosθsinθ dθ∫[0→2sinθ] r³ dr
=(1/4)∫[π/4→π/2] r^4cosθsinθ |[0→2sinθ] dθ
=4∫[π/4→π/2] cosθ(sinθ)^5 dθ
=4∫[π/4→π/2] (sinθ)^5 d(sinθ)
=(2/3)(sinθ)^6 |[π/4→π/2]
=(2/3)(1-1/8)
=7/12
希望可以帮到你,如果解决了问题,请点下面的"选为满意回答"按钮.r指的是区域内的点与原点的距离，不是圆周上的点的范围。