A $60$-ft rope weighing $8$ lb per linear foot is used for mooring a cruise ship. See Figure 54. As the ship prepares to leave port, the rope is released, and it hangs freely over the side of the ship. How much work is done by the deckhand who winds in the rope?

Figure 54

Solution We position an $x$-axis parallel to the side of the ship with the origin $O$ of the axis even with the bottom of the rope and $x=60$ even with the ship's deck. The work done by the deckhand depends on the weight of the rope and the length of rope hanging over the edge.

Partition the interval $[ 0,60]$ into $n$ subintervals, each of length $\Delta x=\dfrac{60}{n}$, and choose a number $u_{i}$ in each subinterval. Now think of the rope as $n$ short segments, each of length $\Delta x.$ Then $$\begin{array}{rcl} \hbox{Weight of the }i{\rm th}\hbox{ segment } &=&8\Delta x\textrm{lb}\\ \hbox{Distance the }i{\rm th}\hbox{ segment is lifted } &=&(60-u_{i}) \textrm{ft}\\ \hbox{Work done in lifting the }i{\rm th}\hbox{ segment } &=&8(60-u_{i}) \Delta x\ \textrm{ft}~\textrm{lb} \qquad {\color{#0066A7}{\hbox{$W=Fx$}}} \end{array}$$

The work $W$ required to lift the $60\textrm{ ft}$ of rope is $$W=\int_{0}^{60}8( 60-x) ~{\it dx}=\big[ 480x-4x^{2}\big]_{0}^{60}=14{,}400~\textrm{ft}~\textrm{lb}$$