Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it.
(a) A heavy rope, 50 ft long, weighs 0.5 lb/ft and hangs over the edge of a building 120 ft high.
(1) How much work is done in pulling the rope to the top of the building?
(2) How much work is done in pulling half the rope to the top of the building?
(b) A cable that weighs 2 lb/ft is used to lift 800 lb of coal up a mine shaft 500 ft deep. Find the work done.
(c) A leaky 10-kg bucket is lifted from the ground to a height of 12 m at a constant speed with a rope that weighs 0.8 kg/m. Initially the bucket contains 36 kg of water, but the water leaks at a constant rate and finishes draining just as the bucket reaches the 12-m level. How much work is done?