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?
- The USGS uses a “midpoint” method to find streamflow. Explain how this works.
- Explain how a “trapezoid rule” and “Simpson’s rule” work for a streamflow measurement.
- In class, we tried this out on a real stream. How did the three methods compare?
- 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?
- Give a detailed explanation as to the sources of errors for these methods. How accurate do you think they are?