I always thought that areas are defined by integrals, until I read Michael Spivak's Calculus p.289:

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> The desire to define area was the motivation, both in this book and historically, for the definition of the integral, but the integral does not really provide the best method of *defining* areas, although it is frequently the proper tool for *computing* them.
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So, what is the best way to define areas? What is the weakness of using integrals to define areas?