化成極坐標形式的積分
x＾2+y＾2=Rx的極坐標方程為r=Rcost (t∈[-π/2,π/2])
又根據對稱性有：
原積分=2∫[0->π/2]∫[0->Rcost] (R^2-r^2)^(1/2)rdrdt
=2∫[0->π/2] -(2/3)(R^2-r^2)^(3/2) | [0->Rcost] dt
=2∫[0->π/2] -(2/3)[(Rsint)^3-R^3] dt
= (4/3)∫[0->π/2] R^3-(Rsint)^3 dt
= (4/3)[R^3(π/2-0) - (R^3)∫[0->π/2] (sint)^3dt]
= (2/3)πR^3-(4/3)(1!/3!)R^3
= (2/3)πR^3-(4/9)R^3
= (2R^3)/3}(π-4/3)
其中用到了∫[0->π/2] (sint)^ndt=(n-1)!/n!當n為奇數時
(π/2)*(n-1)!/n!當n為偶數時
我算出的結果和你給的結果有點出入,也許是我算錯了吧,不過方法就是這樣的