Weight of the bucket is lb/ft.

Weight of the water in the bucket is lb.

Total weight of bucket with water is lb.

Depth of the well is ft.

Bucket with water is pulled up at ft/sec.

Leakage rate of water at the bottom of a hole in bucket is lb/sec.

At any time , consider distance of bucket from bottom of the well as ft.

.

Substitute , and in above expression.

Weight of remaining water in the bucket is lb.

Total weight of the bucket with water at any time is .

Find the work done required to pull the bucket to the top of the well.

Work done as a Riemann sum is  .

lb-ft.

Work done required to pull the bucket to the top of the well lb-ft.