Fluid mechanics of a sinking ball

[Modern Algebr]

A cylinder is full of water, a ball is sinking vertically along the central height the cylinder (ie the centre of the ball is along the central axis of the cylinder) with a constant speed V ,if the radius of the ball is r and the radius of the cylinder is R (r<R), find the volume of water displaced by the ball per unit time

the answer is "Volume displaced per unit time = pi*r^2*V''(area of a circle*V)
whereas I think it should be 2*pi*r^2*V ("area of a semi-spherical*V)

How to imagine this correctly?

Thanks very much for any hints!

 

[Shooting Star]

In one second, the ball travels a distance of V. Think of the “envelop” of the path of the ball through water. If we exclude the initial volume of the ball, then it is a cylinder with a concave hemisphere at the starting end and a convex hemisphere at the other end. The area of cross section of the cylinder is πr². If you cut the convex hemisphere and put it in the concave hollow in the starting end, then it becomes a cylinder with length V and area πr². This is the volume displaced per second.