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In the mathematics and statistics curriculum at Hollins University, we emphasize application of concepts to real life problems. One such application in Calculus II involves the use of integral approximation techniques for computation of the area of the state of Virginia. Students use a "live" on-line map of Virginia (see below) to collect data points and then import these data points into a downloadable Maple 8 worksheet that contains the commands needed (plot, solve, integrate, do loops) for visual and mathematical construction of the area approximations. We choose to use actual worksheets rather than Maplets because students at Hollins are required to read, write and understand Maple code.
Four approximation methods (see below) are presented; they progress from the simpler techniques (rectangular and trapezoidal), which are handled by geometric formula, to a more complex technique (Simpson's Rule), which involves integration of approximating quadratic functions. Instructors may choose to cover the combination of approximation methods that seem most appropriate for their course and students. For example, instructors might assign either the left or right rectangular approximation method, or both, or they might assign only the Simpson's Rule approximation method.
Students are given specific step-by-step instructions for the four area calculations based on the collection of 10 data points. They are then asked to improve their approximations by modifying their data collection and Maple 8 worksheets to include 30 data points. As a project assignment (see below), students are then asked to use these ideas to approximate the area of one of four given states (California, Illinois, Maryland or North Carolina ). This project assignment involves significant and interesting modifications of the Virginia procedure.
Students learn how to mark points on a live map of Virginia.
Students follow eight steps to determine a left rectangular approximation to the area of Virginia. They import 10 data points from the live map of Virginia into a Maple worksheet. This Maple worksheet contains complete Maple code for drawing left rectangular approximations and for determining the area of the approximation based on geometric formulas. As an assignment, students are asked to determine a better approximation using 30 data points. This assignment requires collection of a new set of data points and simple modification of the given Maple worksheet.
Full solution -- view in browser or download as Maple worksheet.
Students follow eight steps to determine a right rectangular approximation to the area of Virginia. Students import 10 data points from the live map of Virginia into a Maple worksheet. This Maple worksheet contains complete Maple code for drawing right rectangular approximations and for determining the area of the approximation based on geometric formulas. As an assignment, students are asked to determine a better approximation using 30 data points. This assignment requires collection of a new set of data points and simple modification of the given Maple worksheet.
Full solution -- view in browser or download as Maple worksheet.
Students follow eight steps to determine a trapezoidal approximation to the area of Virginia. Students import 10 data points from the live map of Virginia into a Maple worksheet. This Maple worksheet contains complete Maple code for drawing trapezoidal approximations and for determining the area of the approximation based on both geometric formulas and simple integration of the approximating linear functions. As an assignment, students are asked to determine a better approximation using 30 data points. This assignment requires collection of a new set of data points and simple modification of the given Maple worksheet.
Full solution -- view in browser or download as Maple worksheet.
Students follow eight steps to determine a Simpson's Rule approximation to the area of Virginia. Students import 10 data points from the live map of Virginia into the given Maple worksheet. This Maple worksheet contains complete Maple code for drawing the Simpson's Rule approximations and for determining the area of the approximation based on both integration of the approximating quadratic functions and the standard Simpson's Rule formula. As an assignment, students are asked to determine a better approximation using 30 data points. This assignment requires collection of a new set of data points and simple modification of the given Maple worksheet.
Full solution -- view in browser or download as Maple worksheet.
In this project assignment, students must approximate the area of California, Illinois, Maryland or North Carolina ("live" maps are provided). These states were chosen so that students must think carefully about how they should modify and extend the previous (Virginia) procedure. For example, the nice horizontal border in these states is the northern (rather than southern) border. Also, each of the four states has some "indentation" in its shape. The end result of the project should be a well-written paper that includes a complete explanation and justification of the student's modified procedure as well as complete Maple worksheets.
For a complete solution, contact phammer@hollins.edu
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