Question

高数格林公式的运用

Answer 1

x² + y² = Rx ==> (x - R/2)² + y² = (R/2)² ==> r = Rcosθ
这是在y轴右边,与y轴相切的圆形
所以角度范围是有- π/2到π/2
又由于被积函数关于x轴对称
由对称性,所以∫∫D = 2∫∫D(上半部分),即角度范围由0到π/2
∫∫ √(R² - x² - y²) dxdy
= ∫∫ √(R² - r²) * r drdθ
= 2∫(0,π/2) dθ ∫(0,Rcosθ) √(R² - r²) * r dr
= 2∫(0,π/2) dθ * (- 1/2) * (2/3)(R² - r²)^(3/2) |(0,Rcosθ)
= (- 2/3)∫(0,π/2) [(R² - R²cos²θ)^(3/2) - R³] dθ
= (- 2/3)∫(0,π/2) R³(sin³θ - 1) dθ
= (- 2/3)R³ * (2!/3!- π/2),这里用了Wallis公式
= (- 2/3)R³ * (2/3 - π/2)
= (1/3)(π - 4/3)R³