Work: The Pool Problem

Do you understand what that formula MEANS? Or where it came from. Frankly, to me it makes no sense. Unless there is an x "hidden" in the E it has the wrong units: the area of a circle is [itex]\pi r^2[/itex], not [itex]\pi r[/itex]!

Imagine dividing the pool into layers- with each layer a specific depth x below the level of the water and thickness [itex]\Delta x[/itex]. That "layer" has area [itex]\pi(11^2)[/itex] square feet and so volume [itex]121\pi \Delta x[/itex] cubic feet. All the water in that "layer" has weight [itex](62.5)121\pi \Delta x[/itex] pounds and it must be raised x+ 5 feet (5 feet up to the water level and another 5 feet to the top of the pool). How much work is required to do that? The total work is found by summing over all the different "[itex]\Delta x[/itex] thickness layers. You turn that Riemann sum into an integral by taking [itex]\Delta x[/itex] smaller and smaller- becoming dx. The limits of integration will be from 0 (the top of the pond) to 4 (the bottom).

 

Can you come up with a mathematical expression describing the amount of work done [tex]\delta W[/tex] for each [tex]\delta V[/tex] infinitesimal volume of water to be raised from its current depth in the pool to its height?