A geometric approach to calculus

Abstract

This investigation explores whether it would be possible to derive the calculus from a geometric basis. The article proves several rules of calculus for polynomials and explains how to differentiate and integrate arbitrary polynomial functions. Differentiation is performed by calculating the slope of a tangent line drawn to a function. Definite integration calculates the area under the graph of a function by comparing this area to a hypersolid's content using Cavalieri's principle.

Keywords:

• calculus
• geometry
• integration
• differentiation
• hypersolid
• Cavalieri's principle

Acknowledgements

I would like to thank my mathematics professor Dr Ken A. Jukes for his very helpful advice and encouragement, as well as for his guidance with the publication process. I would also like to thank my reviewers for their helpful comments.

Notes

Unbeknownst to the author at the time of originally writing this article, the idea of using hyperpyramids to integrate power functions has been used previously by other mathematicians. Nils R. Barth, for example, makes use of this method, although Barth uses a different procedure to calculate the contents of hyperpyramids [4].