Weight of the cable is  lb/ft.

Weight of the coal is  lb.

Depth of the mine is  ft.

Force exerted by the cable to lift it by  ft is  lb. Find the work done needed to lift.the cable to 500 ft.

Therefore, .

Work done as a Riemann sum is  .

.

 lb-ft.

Work done required to lift the cable is  lb-ft.

Find the work done needed to lift.the cable to 500 ft.

Work done formula:

.

Substitute  and  in above expression.

 lb-ft.

Work done required to lift the coal is  lb-ft.

Total work done is sum of the work done needed to lift cable and work done needed to lift the coal.

.

Substitute  and  in above expression.

.

Work done required is  lb-ft. 

Work done required is  lb-ft.