Evaluating Definite Integrals

Discrete and Continuous Perspectives

Abstract

This research project focuses on the construction of summation formulas for five fundamental functions: constant, linear, quadratic, sinusoidal, and exponential. These formulas are derived using arithmetic techniques such as pattern recognition, substitution, and discrete structural analysis. Each summation formula will be evaluated using subintervals of width ∆x = (b−a)/ n , as the number of partition points increases indefinitely, the formulas approach their corresponding definite integrals. This process reveals how these discrete formulas converge to their respective definite integrals without relying on The fundamental theorem of calculus.

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