Question

高等数学三重积分/求问图中第八题用球坐标怎么做？求过程谢谢！

Answer 1

M = ∫∫∫<Ω>ρdv = ∫∫∫<Ω>zdv
= ∫<0,π/2>dφ∫<0,2π>dθ∫<0,2>rcosφ*r^2sinφdr
= ∫<0,π/2>sinφcosφdφ[θ]<0,2π>[r^4/4]<0,2>
= 8π[(sinφ)^2/2]<0,π/2> = 4π.
S<z> = ∫∫∫<Ω>ρzdv = ∫∫∫<Ω>z^2dv
= ∫<0,π/2>dφ∫<0,2π>dθ∫<0,2>(rcosφ)^2*r^2sinφdr
= ∫<0,π/2>sinφ(cosφ)^2dφ[θ]<0,2π>[r^5/5]<0,2>
= (64π/5)[-(cosφ)^3/3]<0,π/2> = 64π/15.
重心立坐标 z = S<z>/M = 16/15
重心坐标 （0, 0, 16/15)

追问

谢谢