How much work is done lifting a leaking bucket from the bottom of a well?

[Precursor]

Homework Statement

A leaky 12 lb bucket is lifted from the bottom of a 10 ft well to the top at
a constant speed with a rope that weighs 0.6 lb/ft. Initially the bucket contains 60 lb
of water but the water leaks at a constant rate and finishes draining just as the bucket
reaches the top of the well. How much work is done?

Homework Equations

w = \int_{a}^{b} [F(x)]dx

The Attempt at a Solution

w = \int_{a}^{b} [F(x)]dx

w = \int_{0}^{10} [0.6(10 - x) + 6(10 - x)]dx

w = 330 lb-ft


Is this correct?

 

[LCKurtz]

Don't forget the weight of the bucket.