In my post Riemann Clouds Improved I showed examples of clouds of values of Riemann sums in such a way that you could see the convergence to the value, the efficiency of the midpoint rule, and other things. Here I include two Riemann sums that are shown
- as manipulable graphs,
- in clouds in an animated form.
To see and work with these demos, you must have Wolfram CDF Player installed on your computer. It is available free from the Wolfram website.
Each manipulable graph (see Elaborate Riemann Sums Demo) has a slider to choose the mesh (1/n) of the partitions. The small plus sign besides the slider gives you additional options. The buttons allow you to choose the type of partition and the type of evaluation points.
Each cloud shows a collection of values of random Riemann sums of the function, plotted by size of mesh (an upper bound on the width of the largest subdivision) and the value of the sum. The cloud shows how the sums converge to the value of the integral.
Every dot represents a random partition. The sums with blue dots have random valuation points, the green dots use the left side of the subdivision, the brown dots the right side, and the red dots the midpoint. The clouds may be suitable for students to study. Some possible questions they could be asked to do are listed at the end.
Pressing the starter shows many clouds in rapid succession. I don't know how much educational value that has but I think it is fun, and fun is worthwhile in itself.
I am not sure of the answers to some of these myself.
- Why is the accuracy generally better for the sine wave than for the quarter circle?
- Why are the green dots above all the others and the brown dots below all the others in the quarter circle?
- Why are they mixed in with the others for the sine curve? In fact why do they tend upward? (Going from right to left, in other words in the direction of more accuracy).
- Why are the midpoint sums so much more accurate?
- Why do they tend downward for the sine wave?
- Is it an optical illusion or do they also tend downward for the quarter circle?
The source code for these demos is Animated Riemann.nb at my Mathematica site.
The animated clouds show two hundred precalculated clouds for each picture, so you get the same clouds each time you run the animation. It would have taken too long to generate the random clouds on the fly. Each list of two hundred took about seven minutes to create on my computer.