Definite Integrals: Riemann Sums to Newton–Leibniz
The Fundamental Theorem of Calculus — from rectangles to the most powerful formula in mathematics
1. The Area Problem & Riemann Sums
1.1 Partitions
Suppose we want the area under a curve $y = f(x) \ge 0$ from $x = a$ to $x = b$. We start by slicing the interval $[a,b]$ into $n$ pieces.
Definition (Partition). A partition of $[a,b]$ is a finite set of points
$$P = \{x_0, x_1, x_2, \ldots, x_n\} \quad\text{with}\quad a = x_0 < x_1 < \cdots < x_n = b.$$
The widths of the sub-intervals are $\Delta x_k = x_k - x_{k-1}$, and the mesh (or norm) of the partition is $\|P\| = \max_k \Delta x_k$.
1.2 Sample Points & Riemann Sums
In each sub-interval $[x_{k-1}, x_k]$ we pick a sample point (or tag) $c_k$. The rectangle over that sub-interval has width $\Delta x_k$ and height $f(c_k)$.
Definition (Riemann Sum).
$$S(f,P,\{c_k\}) = \sum_{k=1}^{n} f(c_k)\,\Delta x_k$$
Common choices for $c_k$:
- Left endpoints: $c_k = x_{k-1}$
- Right endpoints: $c_k = x_k$
- Midpoints: $c_k = \tfrac{x_{k-1}+x_k}{2}$
1.3 Refinement and the Limiting Idea
As we make the mesh $\|P\|$ smaller (equivalently, $n \to \infty$ for equal-width partitions), the rectangles fit the curve more tightly. The Riemann sum approaches a single number — provided that number is the same regardless of how we choose the partition and the sample points.
Definition (Definite Integral). If there exists a number $I$ such that for every $\varepsilon > 0$ there is a $\delta > 0$ with
$$\left|\sum_{k=1}^{n} f(c_k)\,\Delta x_k - I\right| < \varepsilon$$
whenever $\|P\| < \delta$,