Problem: A rural village has no electricity other than a diesel generator for about 2 hours a day. A stream runs nearby, so you (the consulting engineer) are considering the possibility of building a small hydroelectric station on the stream. One piece of information you want is the amount of water flowing in the stream. This is the flow rate of the stream which is given in cubic feet per second. How can you obtain an estimate of the flow rate?
Choose a point along the bank of the stream and imagine a vertical cross-sectional slice through the stream, perpendicular to the direction of flow.
The flow rate (in cu ft/sec) is the amount of water flowing past this cross-section each second. This can be obtained by multiplying the cross-sectional area (in sq. ft.) by the speed (in ft/sec.) at which the water is flowing.
Simplify the problem by assuming that the water flows at the same rate at all points in this cross-section. (This is surely not the case, however; for example, friction and turbulence along the bottom of the stream alter the flow rate there.) You can estimate the flow rate by placing some floats on the surface and timing how long it takes them to travel a given distance.
So now what you need is an estimate of the cross-sectional area. Assume that the stream is about 30 ft wide and no more than 5 ft deep. How would you estimate for the cross-sectional area?
(Keep practical considerations in mind. Your procedure should be feasible and as simple to implement as possible. It should yield sufficiently precise data. Remember that this is an estimate and it is worthwhile to think about or discuss how precise the calculations and measuremenbts need to be.)
Comments: The area of the cross section could be approximated by a set of appropriately chosen rectangles. In calculus, we find the area between an axis and a curve by integration but the underlying concept is the sum of rectangular regions each of whose length is determined by the distance from the curve to the axis.
We have assumed the water flows at the same rate all throughout the cross-section. In fact, along the bottom and sides water flows more slowly than in the center or on the surface. Check out what functions hydrologists use to adjust for friction along the bottom of the stream bed.
How much error is introduced in ignoring this friction?