Let R=sphere radius
Let r=cylinder radius
then sphere Cap height = R-1
considering that V.Sphere.Cap= 1/3*PI*(Cap.Hight)^2*(3*R-Cap.Hight)
=> V.Sphere.Cap = 1/3*PI*(R-1)^2*(3R-(R-1))
V.Sphere.Cap = 1/3*PI*(R^2-2R+1)(2R+1)
V.Sphere.Cap = 1/3*PI*(2R^3-3R^2+1)

V.Ring=Vol.Sphere -2*V.Sphere.Cape-V.cylender
V.Ring=4/3*PI*R^3 - 2*(1/3*PI*(2R^3-3R^2+1)) - 2*PI*r^2
but r^2=R^2 -1^2=R^2-1
substitute on above we get
V.Ring=4/3*PI*R^3 - 2*(1/3*PI*(2R^3-3R^2+1)) - 2*PI*R^2 + 2*PI
V.Ring=4/3*PI*R^3 - 4/3*PI*R^3 + 2*PI*R^2 -2/3*PI - 2*PI*R^2 + 2*PI
V.Ring= -2/3*PI + 2*PI
V.Ring= 4/3 *PI