This sketch is, I hope, an improvement on my previous integration sketch. The concept I tried to capture here is the generality of the Riemann sum construction. The use of Rectangular Approximation Methods LRAM (left), MRAM (midpoint), and RRAM (right) involves the uniform partition of [a,b]. The Riemann sum defines an estimate using an arbitrary partition, which is more eloquent as it does not assume equal sub-intervals. The equal sub-intervals of RAM are not logically optimal, just more easily computed.
For me, this sketch was more of an exercise in the use of dynamic geometry to show generality and randomization. While sketchpad does not contain a random variable calculation, this sketch does utilize a function that affects sufficiently sporadic behavior for the purposes.
There are buttons that move the settings to LRAM, RRAM, MRAM, all with variable numbers of sub-intervals. The function can be changed as well as the interval.
Update: I just updated the file to include three different buttons to quickly change the function. Any of the three functions can be altered individually and the buttons allow quick changes from one function to the next.
Update: (10/23/09) The new version includes a smoother transition between functions when using the action buttons to switch.