题目内容：

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.

优质解答

z = x² + y² + z²
x² + y² + z² - z + 1/4 = 1/4
x² + y² + (z - 1/2)² = (1/2)²
{ x = rsinφcosθ
{ y = rsinφsinθ
{ z = rcosφ
Ω：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