(Answer check) for a work problem

[helloword365]

Homework Statement

A cylinder with radius r = 2 meters and height 4 meters is filled with water up to the top surface of the container. Calculate the minimum amount of energy necessary to remove all the water from the cylinder by pumping it out through the top. (Assume g = 9.8 m/s2 & ρ = 1000 kg/m3)

Relevant Equations

work = force * distance

I talked to other ppl about this problem, and we've all gotten pretty wide-ranging answers, so I was wondering if someone could try and do this so I could see whether my answer is right/wrong. (Answer key does exist but its not that great for this problem).

My work (if needed):

 

[haruspex]

You seem to have treated the vessel as a cone instead of a cylinder.

 

[Lnewqban]

My work (if needed):

What is your calculated number?
Note that the needed sucking force increases as the level of the tank gets lower.

 

[haruspex]

Note that the needed sucking force increases as the level of the tank gets lower.

That is just a different way of analysing it.
@helloword365 considers the work required to lift a parcel of water from a depth of (4m-h) to the surface. The varying force approach considers the force required to lift the whole column of water in the pipe by dy.