请教一道二重积分的题,∫∫x^2ydxdy,D={(x,y)|x^2+y^2

Question

请教一道二重积分的题,
∫∫x^2ydxdy,D={(x,y)|x^2+y^2

Answer 1

转换到极坐标系
D={(x,y)|x²+y²≤4,x≥0, y≥0}
={(r,θ)|r≤2, 0≤θ≤π/2}
∫∫(D)x²ydxdy
=∫∫(D) r²cos²θrsinθrdrdθ
=∫(0,π/2) cos²θsinθdθ ∫(0,2)r^4dr
=-∫(0,π/2) cos²θd(cosθ)*[(r^5)/5]|(0,2)
=-(cos³θ/3)|(0,π/2)*(32/5)
=32/15