Ihave a question about the application of definite integrals. i came up with the following integral:  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

Were is the diagram?