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An alternative way of defining integration in multivariable calculus

Journal Article


Abstract


  • In undergraduate calculus of several variables, double and triple integrals are usually defined as limits of certain Riemann sums. The existence of the integral, as well as the integration formulas, are stated without proof since they require more advanced mathematics. In this article, an alternative and straightforward way of defining multiple integrals is proposed where the usual integration formulas follow as a direct application of the definition. The underlying idea is to map the arbitrary region of integration to an n-dimensional open interval, and integration over the latter is defined via the usual iterated integral. Moreover, the substitution formula is taken as a definition. Numerous illustrative examples are provided.

Publication Date


  • 2022

Citation


  • Rodrigo, M. (2022). An alternative way of defining integration in multivariable calculus. International Journal of Mathematical Education in Science and Technology. doi:10.1080/0020739X.2022.2095540

Scopus Eid


  • 2-s2.0-85133572990

Web Of Science Accession Number


Volume


Issue


Place Of Publication


Abstract


  • In undergraduate calculus of several variables, double and triple integrals are usually defined as limits of certain Riemann sums. The existence of the integral, as well as the integration formulas, are stated without proof since they require more advanced mathematics. In this article, an alternative and straightforward way of defining multiple integrals is proposed where the usual integration formulas follow as a direct application of the definition. The underlying idea is to map the arbitrary region of integration to an n-dimensional open interval, and integration over the latter is defined via the usual iterated integral. Moreover, the substitution formula is taken as a definition. Numerous illustrative examples are provided.

Publication Date


  • 2022

Citation


  • Rodrigo, M. (2022). An alternative way of defining integration in multivariable calculus. International Journal of Mathematical Education in Science and Technology. doi:10.1080/0020739X.2022.2095540

Scopus Eid


  • 2-s2.0-85133572990

Web Of Science Accession Number


Volume


Issue


Place Of Publication