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