Computer Algebra Systems in a Multivariable Calculus Course and Center of Gravity Problems

In teaching a multivariable calculus course, two main difficulties facing the student are to draw surfaces in three-dimensions, and to setup and calculate tedious triple integrals. The added third dimension causes great difficulty even to the well prepared student who has successfully finished a two semester single variable calculus course. The student must now suddenly think in three-dimensions. Visualizing and drawing the corresponding three– dimensional surfaces pose a significant challenge to the novice. To alleviate the problem, the student and the instructor must resort to the modern technologies. Computer algebra systems (CAS) such as Mathematica and Maple are well equipped to handle such tasks. The paper has two goals: One goal is to demonstrate the usage of the CAS Mathematica to learn some standard topics of a multivariable calculus course, such as vectors, partial derivatives, graphing of three-dimensional objects, and multiple integrals. As the second, but most important goal of the paper, we will consider the special topic of the center of gravity of solid objects. This topic was chosen because it uses very many of the techniques learned in a multivariable calculus course. We will show how to use the CAS Mathematica to evaluate tedious triple integrals arising in calculating the center of gravity. Mathematica can also be used as a visualization tool to draw the graphs of three-dimensional solids under consideration. Usually a standard multivariable course only considers the center of gravity of fixed solids. However, with the aid of Mathematica, the students are in an ideal position to consider variable solids as well. Thus, the paper introduces the novel concept of the locus of the center of gravity of certain types of variable solids. The paper also illustrates several facets of A CAS in undergraduate education – the usage of a CAS as a computational tool, visualization tool, experimentation tool, and a conjecture-forming tool.