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Computing integrals over polynomially defined regions and their boundaries in 2 and 3 dimensions

Michael J. Wester
Yuzita Yaacob
Stanly Steinberg
Journal / Anthology

Mathematics and Computers in Simulation
Year: 2011
Volume: 82
Issue: 1
Page range: 79-101

We use 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 Greenís 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 dimensional integration problems in vector calculus textbooks. The three dimensional results are being extended. The results in this paper are being included in an automated student assistant for vector calculus. © 2011 IMACS. Published by Elsevier B.V. All rights reserved.