### 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.

Original language | English |
---|---|

Pages (from-to) | 79-101 |

Number of pages | 23 |

Journal | Mathematics and Computers in Simulation |

Volume | 82 |

Issue number | 1 |

DOIs | |

Publication status | Published - Sep 2011 |

### Fingerprint

### Keywords

- Area integral
- Cylindrical algebraic decomposition
- Iterated integrals
- Line integral
- Surface integral
- Volume integral

### ASJC Scopus subject areas

- Modelling and Simulation
- Numerical Analysis
- Applied Mathematics
- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Mathematics and Computers in Simulation*,

*82*(1), 79-101. https://doi.org/10.1016/j.matcom.2011.06.003

**Computing integrals over polynomially defined regions and their boundaries in 2 and 3 dimensions.** / Wester, Michael J.; Yaacob, Yuzita; Steinberg, Stanly.

Research output: Contribution to journal › Article

*Mathematics and Computers in Simulation*, vol. 82, no. 1, pp. 79-101. https://doi.org/10.1016/j.matcom.2011.06.003

}

TY - JOUR

T1 - Computing integrals over polynomially defined regions and their boundaries in 2 and 3 dimensions

AU - Wester, Michael J.

AU - Yaacob, Yuzita

AU - Steinberg, Stanly

PY - 2011/9

Y1 - 2011/9

N2 - 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.

AB - 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.

KW - Area integral

KW - Cylindrical algebraic decomposition

KW - Iterated integrals

KW - Line integral

KW - Surface integral

KW - Volume integral

UR - http://www.scopus.com/inward/record.url?scp=80054704264&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80054704264&partnerID=8YFLogxK

U2 - 10.1016/j.matcom.2011.06.003

DO - 10.1016/j.matcom.2011.06.003

M3 - Article

AN - SCOPUS:80054704264

VL - 82

SP - 79

EP - 101

JO - Mathematics and Computers in Simulation

JF - Mathematics and Computers in Simulation

SN - 0378-4754

IS - 1

ER -