Abstract: Thousands of written works on the development of calculus over the past two thousand years can be found with a quick google search. These papers and books often either have a very narrow focus, or are incredibly broad in scope, making it difficult to trace how specific concepts changed throughout history. This talk seeks to synthesize the historical development of the elementary definite integral from its origins in Classic Antiquity to our modern definition. We will explore how different mathematicians and philosophers thought about the exact same problems, how differing conceptions can cause rifts between communities, and how a relaxation of rigor can allow for intuitive ideas to blossom; although, of course, it is important to return to rigor in the end.
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