How did Cauchy prove that differentiation and integration are inverse processes?
This suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr2. As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr2—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.
Augustin-Louis, Baron Cauchy provided a rigorous foundation for the fundamental theorem of calculus by shifting the focus of analysis away from geometric or dynamic interpretations toward the concepts of continuity and limit.12
Rather than relying on the algebraic assumptions of his predecessors, Cauchy defined differentiation as the limit of a quotient, , as approaches . He defined the integral independently by approximating the area under a curve using the sum of rectangles, defining the integral as the limit of these sums as the number of rectangles increases indefinitely.1
By establishing these definitions independently of one another, Cauchy was able to rigorously prove that differentiation and integration are mutually inverse processes. This work replaced earlier, less precise methods with a formal framework that remains the basis for modern calculus.1