Projects in Multivariate Calculus

James R. Bozeman, Lyndon State College

In these talks I will present examples of projects assigned each time I teach multivariable calculus. These projects are designed to encourage students to further explore topics which have been covered in the class. The problems also help students to better understand these ideas, and in certain cases lead to lively discussions. I especially try to give projects in which technology, in particular graphing calculators and computers, can be used. The projects that the audience will work on in these presentations are:

Talk 1:  Polar Mysteries

In this project students are asked to graph two polar curves and visually check for points of intersection. They are then asked to solve the two equations simultaneously. The answers to these two questions are different  and the students are asked to completely explain why. This project helps students to see some of the subtleties of polar coordinates as compared to rectangular coordinates.

Talk 2:  Triple Integrals In The Fourth Dimension

In this project students are asked to use triple integrals to find the hypervolume of the 4-dimensional ball. This problem requires extending the ideas used in finding the volume of the 3-dimensional ball with double integrals. It is necessary to use integral tables to solve the triple integrals in both spherical and cylindrical coordinates, and the students are also required to write the formula in rectangular coordinates. This project helps the class to better understand these coordinate systems and their uses in evaluating integrals. The fact that we are finding a result about 4-dimensions leads to many interesting discussions. The book Flatland is also used as a resource for this investigation.