With the second semester now underway, my AP Calculus AB students began their journey into integral calculus by exploring the Riemann Sum, named for the German mathematician Bernhard Riemann (1826-1866). Herr Riemann formalized a specific application of the method of exhaustion pioneered by the Greeks, which itself evolved over time as Eudoxus improved upon Antiphon’s work from the 5th century B.C.E., which led to Archimedes applying the method using triangles to find the area under a paraboloid. Riemann’s contribution used rectangles to estimate the area under any curve. Three different approaches for computing the Riemann Sum estimate for a function, f(x), are shown below.
While leading my students in their journey, I discovered that our new calculus textbook includes a set of problems that need a program that runs on the TI-84 calculator; however, I did not receive the program with the textbook’s ancillary materials. Without the calculator program, students were unable to work a few of the problems I assigned, at least without an extensive amount of work as the problem called for estimates using up to 100, 500, and even 1,000 rectangles each for Left-hand Rectangular Approximation Method (LRAM), Right-hand Rectangular Approximation Method (RRAM), and Midpoint Rectangular Approximation Method (MRAM). None of my students attempted the problem, even for n=10, since the phrase “using the calculator program” stopped them cold.
In fairness, I did warn them that a few of the problems called for a calculator program that we did not have available to use and that I would create a spreadsheet utility for their use. While it was not tremendously difficult to develop, it did take more than three hours to build a semi-extensible set of linked spreadsheets that accepted inputs for an interval [a, b] and a range of values for n, the number of rectangles to use in the estimate, as well as produced outputs of the LRAM, MRAM, and RRAM estimates for a function, f(x). Users of the spreadsheet will need to edit portions for a new f(x) if they wish to use it to explore other functions.
Under the Hood
To help students understand where the multiple values for LRAM, MRAM, and RRAM depicted in a table in their textbook were computed, I elected to recreate that same table in my first attempt. I chose to display the estimates for the rectangular approximation methods in the same table as used to input the values for n, the number of rectangles used in any one estimate. In order to display the solutions to those methods as outputs, a separate worksheet tab for each estimation method contained the detailed interim computations that lead to each solution.
While the result is a data intensive array of x-values for various numbers of rectangles coupled with a correspondingly intensive array of function values, the details depicted in the spreadsheet offered students a peek under the hood of a calculator program, or other online utility like Wolfram MathWorld’s Riemann Sum applet whereas simply using those utilities provide only the estimates in their final form.
While a few students nodded off during our tour of the spreadsheet’s inner workings today, most seemed to benefit from tracing the path of the inputs through various interim calculations ultimately arriving at the table of approximations, which they compared to the one in their textbook. Whether it helped them better understand the use of numerical analysis, finite methods, or Riemann Sums, I am not sure. I do hope my investment in creating the spreadsheet pays dividends though!