Purpose of the project: Apply numeric integration techniques to a real-world problem.

How do you measure streamflow? The basic idea is simple: it is area times velocity. For example, suppose you had a river that was [latex]20[/latex] feet wide, [latex]3[/latex] feet deep, and had water moving at [latex]2[/latex] feet per second. Then we multiply the [latex]20[/latex] and the [latex]3[/latex] to get an area of [latex]60[/latex] ft[latex]^2[/latex], and the multiply by the [latex]2[/latex] ft per second to get [latex]120[/latex] ft[latex]^3[/latex] per second. This works great if you have a rectangular river where the water moves at a constant velocity. But what if the river is not a rectangle? What if the velocity changes depending on where in the river you are? How can you find the streamflow? And what does this have to do with numeric integration techniques?

1. The USGS uses a “midpoint” method to find streamflow. Explain how this works.
2. Explain how a “trapezoid rule” and “Simpson’s rule” work for a streamflow measurement.
3. In class, we tried this out on a real stream. How did the three methods compare?
4. Try all three methods on the virtual river (link) with [latex]n = 4[/latex] and [latex]n = 8[/latex]. I recommend using a spreadsheet to record your data and to automatically do the calculations for you. How did the methods compare now? How does changing [latex]n[/latex] affect the answers?
5. Give a detailed explanation as to the sources of errors for these methods. How accurate do you think they are?