Calculus Work Problems

[jeffsours]

Hey guys. i've got a couple of really confusing questions on my study guide and i'm testing tomorrow. I was hoping someone could shed some light with these and I really appreciate the help so thanks

1.
[h=2]Find the work in ft-lbs required to empty a right cylindrical tank with a radius of 8 ft, a height of 6 ft, and a water level of 4 ft by pumping the water to the top of the tank[/h]2.

Find the work (in ft-lbs) required to empty all but the bottom 2 feet of a full right cylindrical tank with a radius of 5 ft and a height of 10 ft by pumping the water to 1 foot above the top of the tank

again thanks for all the help.

 

[HallsofIvy]

Imagine a thin "layer", of thickness "dx", of water at height "x" above the bottom of the barrel. Since the radius is 8 feet, the area of that layer is \(\displaystyle \pi (8)^2= 64\pi\) square feet and has volume \(\displaystyle 64\pi dx\) cubic feet. I presume you are given the density of water or can look it up. Taking that density to be \(\displaystyle \delta\) the weight of that layer of water is \(\displaystyle 64\pi\delta dx\) pounds. It has to be lifted a distance 6- x ft so, since work is "force times distance", the work required to lift that layer is \(\displaystyle 64\pi\delta (6- x)dx\). Since the water goes only up to 4 feet, integrate from 0 to 4: \(\displaystyle 64\pi\delta\int_0^4 (6- x)dx\)

 

[jeffsours]

So, 1024pi is our final after finding the integral, correct?