What concept describes the definite integral as the limit of a sum?

The concept that describes the definite integral as the limit of a sum is the Riemann Sum. This approach is foundational in understanding how integration is defined in calculus.

Riemann Sums break down the area under a curve into a finite number of rectangles, each representing a segment of the area. As the number of rectangles increases (and their width decreases), this sum approaches a precise value, known as the definite integral. Formally, if a function is continuous on a closed interval, the limit of the Riemann Sum as the number of rectangles approaches infinity yields the exact value of the definite integral.

This concept is crucial for understanding the approximation of areas under curves and serves as the basis for many numerical integration techniques. Using Riemann Sums allows students to visualize integration as an accumulation of infinitesimally small quantities, bridging the gap between discrete and continuous mathematics.