Definite Integral as the Limit of a Sum

This section makes precise the informal picture used above — an area built by summing infinitely many thin strips — as the formal definition of the definite integral. Two sub-topics follow: §9.2.1 defines the Riemann sum, the Riemann integral, and the three standard rules (left-end, right-end, mid-point) for approximating it from a finite partition; §9.2.2 specialises to equal-width partitions and derives the closed limit formula $\int_a^b f(x)\,dx=\lim_{n\to\infty}\frac{b-a}{n}\sum_{r=1}^n f\!\left(a+\frac{(b-a)r}{n}\right)$, the working tool used to evaluate a definite integral directly "as the limit of a sum" ( …