How did Pierre de Fermat’s methods for areas anticipate integral calculus?

Pierre de Fermat’s work in the mid-1630s significantly anticipated integral calculus through his development of a systematic summation process for calculating areas bounded by curves.1

By generalizing equations for various curves—such as parabolas and hyperbolas of the form $a^{n-1}y=x^n$—Fermat derived formulas for the areas under these curves. His approach utilized a summation process that is mathematically equivalent to the modern formula used in integral calculus for finding the area under a curve. Specifically, he employed ingenious transformations to handle complex algebraic curves, allowing him to determine areas that were previously considered difficult or impossible to compute.1

While Fermat successfully applied these methods to calculate areas, find centers of gravity, and determine the lengths of curves, it is not known if he recognized that his integration process was the inverse of his method for finding tangents (which anticipated differential calculus). His work in this area, along with his contributions to analytic geometry, established him as a foundational figure in the development of modern mathematical analysis.1