Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A cable that weighs 2 lb/ft is used to lift 800 lb of coal up a mine shaft 500 ft deep. Find the work done.
The work done
Approximate the required work by using the concept of Riemann sum. Then express the work as an integral and evaluate it.
i) Weight of the cable
ii) Length of the cable
3) Definition 3:
Let’s place the origin at the top and -axis pointing downward as in the figure.
Divide the cable into small parts with length
If is a point in the such interval, then all points in the interval are lifted by approximately the same amount, namely,
The cable weighs pound per foot
Therefore, the weight of part is
By using definition 3,
Thus, the work done on the part, in foot-pounds, is
The total work done by adding all these approximations and letting the number of parts become large (so )