不用球面坐标,解这三题

11.
I = ∫∫∫Ω √[1 - (x² + y² + z²)^(3/2)] dv
= ∫∫∫Ω √(1 - r³) * sinφ * r² drdφdθ
= ∫(0,2π) dθ ∫(0,π) sinφ dφ ∫(0,1) √(1 - r³) * r² dr
= 2π * 2 * (- 1/3)(2/3)(1 - r³)^(3/2)：(0,1)
= 4π * (- 2/9) * (- 1) * 1
= 8π/9
12.
I = ∫∫∫Ω (x + z) dxdydz.底部是圆锥z = √(x² + y²),顶部是球体z = √(1 - x² - y²)
√(x² + y²) = √(1 - x² - y²) ==> 2x² + 2y² = 1 ==> x² + y² = 1/2
积分域在xoy面对称,x是奇函数,该积分等于0.
I = ∫∫∫Ω z dxdydz
= ∫(0,2π) dθ ∫(0,1/√2) r dr ∫(r,√(1 - r²)) z dz
= 2π * ∫(0,1/√2) (1/2 - r²) * r dr
= 2π * 1/16
= π/8
17.
z² = h²/R² * (x² + y²) ==> x² + y² = (Rz/h)² ==> Dz的面积 = π * R²z²/h²
底部是圆锥z² = h²/R² * (x² + y²),顶部是平面z = h
I = ∫∫∫Ω z dxdydz
= ∫(0,h) z (∫∫Dz dxdy) dz
= ∫(0,h) z * (Dz的面积) dz
= ∫(0,h) z * π * R²z²/h² dz
= πR²/h² * ∫(0,h) z³ dz
= πR²/h² * h⁴/4
= (1/4)π(Rh)²