高等数学三重积分? 50
3个回答
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采用柱坐标
I = ∫∫∫<Ω>e^zdv = ∫<1, 2>e^zdz∫<0, 2π>dt∫<0, z>rdr
= 2π∫<1, 2>e^zdz[r^2/2]<0, z> = π∫<1, 2>z^2e^zdz
∫z^2e^zdz = ∫z^2de^z = z^2e^z-∫2ze^zdz
= z^2e^z-∫2zde^z = z^2e^z-2ze^z+2e^z = (z^2-2z+2)e^z
I = π[(z^2-2z+2)e^z]<1, 2> = π(2e^2-e) = πe(2e-1)
I = ∫∫∫<Ω>e^zdv = ∫<1, 2>e^zdz∫<0, 2π>dt∫<0, z>rdr
= 2π∫<1, 2>e^zdz[r^2/2]<0, z> = π∫<1, 2>z^2e^zdz
∫z^2e^zdz = ∫z^2de^z = z^2e^z-∫2ze^zdz
= z^2e^z-∫2zde^z = z^2e^z-2ze^z+2e^z = (z^2-2z+2)e^z
I = π[(z^2-2z+2)e^z]<1, 2> = π(2e^2-e) = πe(2e-1)
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