Question

计算三重积分|x|+y+|z|dv,怎么计算

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Answer 1

利用对称性可以去掉绝对值，然后用极坐标

Answer 2

标准球坐标 x2+y2+(z-a)2 = a2 x2+y2+z2 = 2az x = r sinφ cosθ y = r sinφ sinθ z = r cosφ dV = r2sinφ drdφdθ Ω方程变为：r = 2acosφ 由于整个球面在xOy面上，所以0 ≤ φ ≤ π/2 ∫_(Ω) (x2+y2+z2) dV = ∫(0,2π) dθ ∫(0,π/2) sinφ dφ ∫(0,2acosφ) r2 * r2 dr = (2π)∫(0,π/2) sinφ * (1/5)(32a?cos?φ) dφ = (2π)(1/5)(32a?)(- 1)∫(0,π/2) cos?φ d(cosφ) = (2π)(1/5)(32a?)(- 1)(1/6)[ cos?φ ]|(0,π/2) = (2π)(1/5)(32a?)(- 1)(1/6)(0 - 1) = 32πa?/15