COMPUTING INTEGRALS OVER POLYNOMIALLY DEFINED REGIONS AND THEIR BOUNDARIES IN 2 AND 3 DIMENSIONS

Abstract

This study uses the cylindrical algebraic decomposition algorithms implemented in Mathematica to produce procedures to analytically compute integrals over polynomially defined regions and their boundaries in two and three dimensions. Using these results, we can implement the divergence theorem in three dimensions or the Green’s theorems in two dimensions. These theorems are of central importance in the applications of multidimensional integration. They also provide a strong correctness test for the implementation of our results in a computer algebra system. The resulting software can solve many of the two and some of the three dimension a lintegration problems in vector calculu stextbooks. The three dimensional results are being extended. The results in this paper are being included inanautomated student assistant for vector calculus.

Keywords

• d: area integral
• line integral
• volume integral
• iterated integrals
• cylindrical algebraic decomposition (cad)