A 100 ft rope, weighing 0.1 lb/ft, hangs over the edge of a tall building.
\[ 0 \leq x \leq 100\\ dW = \text{amount of work required to lift a segment of the rope a distance x from its current position} =0.1 x dx\\ x = \text{height}\\ W = \text{Work} = \int_0^{100} dW = \int_0^{100} 0.1 x dx\\ = \int_0^{100} 0.1 x dx = \frac{0.1x^2}{2} |_0^{100} = 0.05x^2 |_0^{100}\\ = 0.05(100)^2 - 0.05(0)^2 = 0.05(100)^2 = 500 \text{ft-lbs} \] 500 ft-lbs of work is done pulling the entire rope to the top of the building.
\[ \text{Half the total work: }\\ \frac{1}{2} W = \frac{1}{2} \cdot 500 = 250 \text{ft-lbs}\\ \] We need to find the height \(y\) where pulling the rope from \(y\) feet to the top results in 250 ft-lbs of work.
\[ \int_y^{100} 0.1 x dx = 250\\ \frac{0.1x^2}{2} |_y^{100} = 250\\ 0.05(100)^2 - 0.05(y^2) = 250\\ 0.05y^2 = 250\\ y^2 = 5000\\ y = 70.71 \text{ ft} \] 70.71 feet of rope is pulled when half the total work is done.