Question

计算三重积分

∫∫∫xycoszdxdydz 区域为抛物柱面y=根号下x,与平面y=0,z=0,x+z=π/2所围成的区域

Answer 1

积分区域为：0≤x≤π/2，0≤y≤√x，0≤z≤π/2-x
∫∫∫xycoszdxdydz
=∫<0,π/2>x{∫<0,√x>ydy*∫<0,π/2-x>coszdz}dx
=∫<0,π/2>x{[<0,√x>y^2/2]*[<0,π/2-x>-sinz]}dx
=∫<0,π/2>x{x/2*(-cosx)}dx
=-∫<0,π/2>x^2/2dsinx
=-[<0,π/2>x^2/2*sinx]+∫<0,π/2>sinxd(x^2/2)
=-π^2/8+∫<0,π/2>xsinxdx
=-π^2/8-∫<0,π/2>xdcosx
=-π^2/8-[<0,π/2>xcosx]+∫<0,π/2>cosxdx
=-π^2/8-0+∫<0,π/2>dcosx
=-π^2/8+[<0,π/2>sinx]
=-π^2/8+1

追问

答案是错的o(╯□╰)o