高数一道三重积分的题目,有图求解
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化为柱坐标。x^2+y^2 = r^2, x^2+y^2 = z^2, z = r
I = ∫∫∫<Ω>√(x^2+y^2)dxdydz
= ∫<0, 1>dz∫<0, 2π>dt∫<0, z>r rdr
= 2π∫<0, 1>dz[r^3/3]<0, z>
= (2π/3)∫<0, 1>z^3dz
= (2π/3)[z^4/4]<0, 1> = π/6
I = ∫∫∫<Ω>√(x^2+y^2)dxdydz
= ∫<0, 1>dz∫<0, 2π>dt∫<0, z>r rdr
= 2π∫<0, 1>dz[r^3/3]<0, z>
= (2π/3)∫<0, 1>z^3dz
= (2π/3)[z^4/4]<0, 1> = π/6
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