求xy的二重积分，其积分区域D为x^2+y^2=2y,y=x与y轴的构成的封闭图形

Question

如题所述

Answer 1

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

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