Calculus
posted by Mason .
An electric elevator cable with a motor at the top has a multistrand cable weighing 4.5lb/ft. When the car is at the first floor, 180ft of cable are pulled out, and effectively 0ft are out when the car is at the top floor. How much work does the motor do just lifting the cable when it takes the car from the first floor to stop.

work = integral w dz
w = 4.5 (180z)
so
work = integral from z = 0 to z = 180 of 4.5 (180z)dz
= 4.5*180 (180)  4.5 (1/2)(180)^2
= (1/2)(4.5)(180)(180)
or half the weight over the whole distance of course :)