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.