# More Integrals and work

1. Mar 11, 2007

### Jacobpm64

1. The problem statement, all variables and given/known data

Water in a cylinder of height 10 ft and radius 4 ft is to be pumped out. Find the work required if

(a) The tank is full of water and the water is to be pumped over the top of the tank.

(b) The tank is full of water and the water must be pumped to a height 5 ft above the top of the tank.

(c) The depth of water in the tank is 8 ft and the water must be pumped over the top of the tank.

2. Relevant equations

W = F * D
Density of water = 62.4 lb/ft^3 (weight)

3. The attempt at a solution
I know that I have to slice up the cylinder into arbitrarily small cylinders and find the work for each cylinder. I'm not sure how to slice it, and the thing that really confuses me is how the limits of integration change with each problem.

2. Mar 12, 2007

### HallsofIvy

Staff Emeritus
Your problem is that distance each "piece of water" has to be lifted varies with its height in the tank. "Slice" the water to get as large as possible, all at the same height. How many ways are there to slice a cylindrical tub into cylinders any way?

Take a thin "layer" of water at height "y", thickness "dy". What is its area? What is its volume? What is its weight? (Those will be the same for all y and each problem a, b, c.) What height does the layer of water have to be lifted? What work has to be done to lift the layer of water? (Those will depend on y and will be different for a and b.)

Adding those gives a Riemann sum that approximates the work done in lifting all of the water. Convert it to an integral. In c, the limits of integration will be different than in a or b.