计算三重积分 5
∫∫∫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...
∫∫∫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
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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
∫∫∫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
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