Limit Definition

The Limit Definition of the Definite Integral

Finding the exact area under a curve requires a new approach: we approximate using rectangles, then take a limit to make the approximation exact.

Building the Riemann Sum

Divide the interval $[a,b]$ into $n$ equal subintervals of width $\Delta x = \frac{b-a}{n}$. In each subinterval, choose a sample point $x_i^*$ and form a rectangle with height $f(x_i^*)$ and width $\Delta x$. The Riemann sum $\sum_{i=1}^{n} f(x_i^*)\,\Delta x$ approximates the total area. As $n$ increases, the rectangles fit the curve more closely and the approximation improves.

The Limit: Definition of the Definite Integral

As $n \to \infty$, the width $\Delta x \to 0$, and the Riemann sum converges to the exact signed area. The definite integral is defined as this limit: $\int_a^b f(x)\,dx = \lim_{n\to\infty} \sum_{i=1}^{n} f(x_i^*)\,\Delta x$

This limit gives the signed area: positive where $f(x) \geq 0$ and negative where $f(x) < 0$. Remarkably, for continuous functions, this limit exists regardless of how the sample points $x_i^*$ are chosen—left endpoints, right endpoints, midpoints, or any other point within each subinterval all yield the same result.

Why This Matters

This limit-based definition is foundational. It transforms a geometric problem (finding area) into a rigorous mathematical definition that works for any continuous function. More importantly, it connects to the Fundamental Theorem of Calculus, which provides a practical shortcut: instead of computing the limit directly, we can evaluate an antiderivative.