作柱面坐标变换,设x=rcosφ,y=rsinφ,z=z
故∫∫∫|z-x^2+y^2|dxdydz
=∫(0,2π)dφ∫(0,√2)rdr∫(0,1)|z-r|dz (符号∫(a,b)表示从a到b积分,以下类同)
=2π[∫(0,1)rdr∫(r,1)(z-r)dz+∫(0,1)rdr∫(0,r)(r-z)dz+∫(1,√2)rdr∫(0,1)(r-z)dz]
=2π[∫(0,1)(r/2-r²+r³/2)dr+∫(0,1)(r³/2)dr+∫(1,√2)(r²-r/2)dr]
=2π(1/24+1/8+2√2/3-7/12)
=2π(8√2-5)/12
=π(8√2-5)/6.