Math  Calculus 2
posted by Amy .
An underground tank full of water has the following shape:
Hemisphere  5 m radius. at the bottom
Cylinder  radius 5 m and height 10m in the middle
Circular cone radius 5 m and height 4 m at the top
The top of the tank is 2 m below the ground surface and is connected to the surface by a spout. find the work required to empty the tank by pumping all of the water out of the tank up to the surface.
density of water = 1000 kg/m^3
Gravity = 10 m/s^2

There are at least two ways to do this.
You can calculate the distance below the ground of the center of gravity of the water in the tank. You are then lifting the mass of water from that point to the ground and work = m g times that distance up.
The other way is to find the cross sectional area of every slice of the tank at distance below ground z, call it Az. The work done on a slice is then rho g Az dz . Integrate that from z = 0 to z = 16. 
The work done on a slice is then rho g Az z dz . Integrate that from z = 0 to z = 16.
(forgot distance slice is lifted, z)