Formal Definition of the Definite Integral

The formal definition of the definite integral is a fundamental concept in calculus, providing a precise mathematical framework for calculating the accumulated total of a function over a specified interval. Developed by mathematicians such as Isaac Newton and Gottfried Wilhelm Leibniz, this definition involves partitioning the interval, evaluating the function at specific points within each subinterval, and taking the limit as the partition becomes infinitesimally small. The result is a powerful tool for analyzing and solving problems related to area, displacement, and other applications in mathematics and physics.