Calculus- Finding the Work Required to Fill a Tank Example 1

A common level 2 calculus question is to ask for the work required to move a liquid a certain distance, or into a tank of some common geometric shape. Sometimes the tank is spherical, conical, or cylindrical.

In the problem below we explore a question involving a spherical tank.
The center of the tank is elevated to a height of 150 meters from the ground.  The radius of the tank is 25 meters.
We are asked to compute the work done by filling the tank up to a level of 125 meters with homogeneous water. 

Note:
-Remember that work depends on initial and final state of the water or liquid.
-The weight of each 'dy' layer of water is different and the distance each 'dy' layer must be moved is also different.
-Our strategy is to compute the work required to move one tiny layer, then use our knowledge of integration to sum all the layers together.
-The problem assumes that the liquid being moved is homogeneous, this means that it is of uniform density throughout. 
-Be certain of the interval over which you are integrating.  Sometimes they ask to pump the water out of the tank through the top rather than into the tank through the bottom.
-In this problem we need only integrate from 100 meters to 125 meters since that is the height the water must be pumped to.