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

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  • Wester, Michael J.
  • Yaacob, Yuzita
  • Steinberg, Stanly

Abstract

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 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 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.

Suggested Citation

  • Wester, Michael J. & Yaacob, Yuzita & Steinberg, Stanly, 2011. "Computing integrals over polynomially defined regions and their boundaries in 2 and 3 dimensions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(1), pages 79-101.
    • Handle: RePEc:eee:matcom:v:82:y:2011:i:1:p:79-101
    DOI: 10.1016/j.matcom.2011.06.003
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    References listed on IDEAS

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    1. Sawada, Hiroyuki & Yan, Xiu-Tian, 2004. "Application of Gröbner bases and quantifier elimination for insightful engineering design," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 67(1), pages 135-148.
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