化成极坐标形式的积分
x^2+y^2=Rx的极坐标方程为r=Rcost(t∈[-π/2,π/2])
又根据对称性有:
原积分=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)^3dt
=(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!当n为奇数时
(π/2)*(n-1)!/n!当n为偶数时
我算出的结果和你给的结果有点出入,也许是我算错了吧,不过方法就是这样的