### Meg Wynne

*Pronouns: She/Her/Hers*

**UROP Fellowship**: Mcubed Scholars

**Research Mentor(s)**: Robert Krasny, PhD, Arthur F. Thurnau Professor of Mathematics

Mathematics

Presentation Date: Wednesday, July 29, 2020 | Session 2 | Presenter: 4

**Authors**: Meg Wynne, Robert Krasny

#### Abstract

The Fundamental Theorem of Calculus is one of the crowning achievements in mathematics and other scientific disciplines. The theorem yielded one of the most important ideas of modern math – the evaluation of the integral using antiderivatives. In my summer research I study two sides of the integral – the theoretical component as well as the computational and numerical methods that allow for its implementation into applied mathematics as well as other fields. When studying the theoretical side we can start to find relationships between finite sums and certain functions, which gives us information on the rules surrounding the integration methods of certain functions. This can mean the difference between an exact integral and an approximate value that was computed using numerical techniques. In our efforts to prove relationships between sums and integrals we also found various recursive relationships between sums and geometry, which provide a geometrical intuition into numerical relationships as well as a pattern to help us understand area or volume through sums. On the numerical and computational side of integrals, we are able to determine which summation techniques are exact for polynomials of differing degrees and trigonometric functions. One of the most important aspects of these different summation techniques is their error and how their accuracy can be tweaked to discover new summation methods. We find, for example, that although the Trapezoid Rule and the Midpoint Riemann Sum are accurate for the same polynomial degree, their error can behave unexpectedly when they are used to approximate the integral of various functions. We study how this information can be useful when deciding which summation rule to use and why. Our final byproduct in the exploration of the integral was the investigation of rational numbers as they are used to approximate irrational numbers. Many methods, such as Taylor Series, were used as both approximation tools for irrational numbers as well as approximation tools for both local and global error analysis.

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Presentation Script

#### Research Disciplines

Social Sciences