Work: The Pool Problem

[RedBarchetta]

Homework Statement

A circular swimming pool has a diameter of 22 ft, the sides are 5 ft high, and the depth of the water is 4 ft. How much work is required to pump all of the water over the side? (Use the fact that water weighs 62.5 lb/ft^3.)

(a) Show how to approximate the required work by a Riemann sum. Give your answer using the form below. (If you need to enter -infinity or infinity, type -INFINITY or INFINITY.)
http://www.webassign.net/www21/symImages/5/b/9d364b30bb301d2a03a8ec803e5bf6.gif

(b)Express the work as an integral and evaluate it.

The Attempt at a Solution

I've found everything in the riemann sum equation minus the "E". What would this be?
A=N
B=infinity
C=1
D=N
E=?

One I find that, I should be able to setup the integral and solve it.

Thanks!

 

[HallsofIvy]

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 $\pi r^2$, not $\pi r$!

Imagine dividing the pool into layers- with each layer a specific depth x below the level of the water and thickness $\Delta x$. That "layer" has area $\pi(11^2)$ square feet and so volume $121\pi \Delta x$ cubic feet. All the water in that "layer" has weight $(62.5)121\pi \Delta x$ 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 "$\Delta x$ thickness layers. You turn that Riemann sum into an integral by taking $\Delta x$ smaller and smaller- becoming dx. The limits of integration will be from 0 (the top of the pond) to 4 (the bottom).

 

[Defennder]

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

 

[RedBarchetta]

Thanks HallsOfIvy & Defennder. Well, now I've solved the problem and found the amount of work, but the elusive "E" variable in that question escapes me. :rofl: