A heavy rope, 40 feet long, weighs 0.3 lb/ft and hangs over theedge of a building 110 feet high. Let x be the distance in feetbelow the top of the building.
Find the work required to pull the entire rope up to the top of thebuilding.

1. Draw a sketch of the situation.
We can look at this problem two different ways. In either case, wewill start by thinking of approximating the amount of work done byusing Riemann sums. First, let’s imagine “constant force changingdistance.”

2. Imagine chopping the rope up into n pieces of length ?x. Howmuch does each little
piece weigh? (This is the force on that piece of rope. It should bethe SAME for each
piece of rope.)
3. How far does the piece of the rope located at xi have to travelto get to the top of the
building? (Notice that this is DIFFERENT for each piece of rope; itdepends on the
location of the piece.)
4. How much work is done (approximately) to move one piece of ropeto the top of the
building?
5. Find the amount of work required to pull the entire rope to thetop of the building,
using an integral.

Now, let’s do the same problem, but this time, imagining “constantdistance, changing force.” Imagine pulling up the rope a little bitat a time, say, we pull up ?x feet of rope with each pull.
6. After you have pulled up xi feet of rope, how much of the roperemains to be pulled?
7. What is the force on the remaining amount of rope?
8. Remembering that each pull moves the rope ?x feet, how much workis done for each
pull?
9. Find the amount of work required to pull the entire rope to thetop of the building,
using an integral.

Help 6 to 9 and write it in order with number.