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.