Riemann Sums, Summation Notation, and Definite Integral Notation

Start with a Riemann sum: slice [a, b] into n pieces, each of width delta x, pick a point x_i in each piece, and add up f(x_i) times delta x over all i. That sum is a finite approximation to the area under the curve, and it's only as good as how thin your slices are. Make n huge and delta x tiny, and the jagged rectangle-top approximation hugs the curve almost perfectly. The definite integral is defined as exactly what that sum approaches as n goes to infinity and delta x goes to 0 — not an approximation anymore, but the exact area, signed by whether f is above or below the x-axis.

The notation change tracks the conceptual change. Sigma with index i running from 1 to n becomes the integral sign, which is itself a stretched S marking a sum — but one taken in the limit. The finite delta x, a real nonzero width, becomes dx, representing an infinitesimal width once n is infinite. The discrete sample point x_i becomes the continuous variable x sweeping through [a, b]. Every piece of integral notation is a fossil of the Riemann sum it came from.

This is the hinge between the unit's buildup and everything after it. Once you accept that the integral sign means 'limit of a Riemann sum,' the Fundamental Theorem of Calculus is the shortcut that lets you skip ever computing that limit directly, using antiderivatives instead. Riemann sums also return later whenever a function isn't easily antidifferentiated, or when you build integrals from tables of data instead of formulas.