计算三重积分∫∫∫(x^2+y^2+z^2)dv,其中Ω由z=x^2+y^2+z^2所围成的闭区域.

z = x² + y² + z²x² + y² + z² - z + 1/4 = 1/4x² + y² + (z - 1/2)² = (1/2)²{ x = rsinφcosθ{ y = rsinφsinθ{ z = rcosφΩ：r² = rcosφ → r = cosφ∫∫∫ (x² + y² + z²) dV= ∫∫∫ r² * r²sinφ dV = ∫∫∫ r⁴sinφ dV= ∫(0→2π) ∫(0→π/2) ∫(0→cosφ) r⁴sinφ drdφdθ= 2π ∫(0→π/2) sinφ * (1/5)r⁵：(0→cosφ) dφ= 2π/5 ∫(0→π/2) cos⁵φsinφ dφ= - 2π/5 ∫(0→π/2) cos⁵φ d(cosφ)= - 2π/5 * (1/6)cosφ：[0→π/2]= - π/15 * (0 - 1)= π/15