I have a question about the application of definite integrals. I came up with the following integral:[tex] \int\limits^5_0 {62.5 * 6 * 2 * x} \, dx [/tex]This is equal to 9780. Unfortunately, 9780 ft-lb is not an answer, but 9375.0 ft-lb is! Can someone tell me what is wrong with my integral and if 9375.0 ft-lb is the answer to my question?

You have the correct integral set up. I'm not sure how you're getting 9780. The correct answer is 9375.0 foot-pounds

The volume of each slice is 6*2*DeltaX cubic feet. So the force pulling down on each slice is 62.5*6*2*DeltaX pounds as this is the weight of the slice. 

The work done to move that slice x feet upward is 62.5*6*2*DeltaX*x or 62.5*6*2x*DeltaX as you wrote for your integrand.

Integrating with respect to x will effectively add up all the horizontal slices to get the total work done

See the attached image for the integral and steps leading to the answer of 9375