Question

求球体x^2+y^2+z^2<=4R^2被圆柱面x^2+y^2=2Rx(R>0)所截得的（含在圆柱面内部分）立体的体积。谢啦！

Answer 1

用柱坐标替换x=rcosθ，y=rsinθ,z=z.则球体方程变为：r^2+z^2=4R^2,柱面方程变为r=2Rcosθ.则对于上半段有：-π/2<θ<π/2,0<r<2Rcosθ,0<z<根号（4R^2-r^2）体积=
2∫dθ∫dr∫rdz=16/3*R^3(π-4/3)