∫∫∫（G）（x^2+y^2）dv,其中G为旋转抛物面z=1/2(x^2+y^2)与平面z=3所围成求三重积分详细过程

{ z = 3、在上方
{ 2z = x² + y²、在下方
柱坐标(投影法)：2z = x² + y² --> 2z = r²、x² + y² = 2(3) = 6 --> r² = 6 --> 0 ≤ r ≤ √6
∫∫∫(G) (x² + y²) dV
= ∫∫(Dxy) dxdy ∫(r²/2~3) r² dz
= ∫(0~2π) dθ ∫(0~√6) r dr ∫(r²/2~3) r² dz
= 2π • ∫(0~√6) r³ • (3 - r²/2) dr
= π • ∫(0~√6) (6r³ - r⁵) dr
= π • [ (6/4)r⁴ - (1/6)r⁶ ] |(0~√6)
= π • [ (3/2)(√6)⁴ - (1/6)(√6)⁶ ]
= 18π
柱坐标(切片法)：x² + y² = 2z --> x² + y² = (√2√z)² --> 0 ≤ r ≤ √(2z)
∫∫∫(G) (x² + y²) dV
= ∫(0~3) dz ∫∫(Dz) (x² + y²) dxdy
= ∫(0~3) dz ∫(0~2π) dθ ∫(0~√(2z)) r² • r dr
= ∫(0~3) dz • 2π • (1/4)[ r⁴ ] |(0~√(2z))
= (π/2)∫(0~3) 4z² dz
= 2π • (1/3)[ z³ ] |(0~3)
= 2π • (1/3)(27)
= 18π