This is not really a homework problem but just a question.

How come while doing pump problems distance is part of the equation but rope problems distance is not...

For example your rope integral might look like integral from zero to fifty of (25-(.5)x) dx

Here all that 25-(.5)x is the weight of the rope...

But a pump problem may look like the integral from zero to three of ((8x)(9.8)(1000)(5-x))dx

Here (5-x) represents the distance the water travels...

How come there is nothing that represents the distance the rope travels in the rope problem?