Question

高数 积分3

Answer 1

答：(3π - 4)R³/18

为了方便被积函数化简，运用标准极坐标换元
x² + y² = Rx，y ≥ 0为上半圆
r = R cosθ，0 ≤ θ ≤ π/2
∫∫_(D) √(R² - x² - y²) dxdy
= ∫(0,π/2) dθ ∫(0,Rcosθ) √(R² - r²) * r dr
= ∫(0,π/2) (- 1/2)(2/3)(R² - r²)^(3/2) |(0,Rcosθ) dθ
= (- 1/3)∫(0,π/2) [ (R² - R²cos²θ)^(3/2) - (R²)^(3/2) ] dθ
= (- 1/3)∫(0,π/2) R³( |sinθ|³ - 1 ) dθ，在第一象限sinθ ≥ 0，故去掉绝对值
= (R³/3)∫(0,π/2) ( 1 - sin³θ) dθ，运用Wallis公式化简后面那项
= (R³/3)(π/2 - 2/3)
= (3π - 4)R³/18