This central tenet of calculus establishes a profound connection between the two main operations: differentiation and integration. In essence, it states that the definite integral of a function can be evaluated by finding an antiderivative of that function and evaluating it at the limits of integration. As a practical illustration, consider determining the area under a curve described by a function. This area can be found directly by finding an antiderivative of the function and subtracting its value at the lower limit of integration from its value at the upper limit.
The significance of this principle lies in its ability to simplify the calculation of definite integrals. Without this foundational theorem, evaluating definite integrals would often require complex limit processes and summation techniques. Furthermore, it provides a theoretical basis for many practical applications of calculus in various fields, including physics, engineering, and economics. Historically, the formalization of this relationship was a culmination of efforts by mathematicians such as Isaac Newton and Gottfried Wilhelm Leibniz, solidifying calculus as a powerful and cohesive mathematical discipline.
