The Definite Integral
Big Idea
The definite integral answers one of the most fundamental questions in mathematics: How do we find the exact area under a curve? It connects the idea of summing infinitely many infinitely thin rectangles to a precise numerical value.
1. From Riemann Sums to the Definite Integral
Recall that we can approximate the area under a curve $f(x)$ from $x = a$ to $x = b$ by dividing the interval into $n$ rectangles. Each rectangle has:
- Width: $\Delta x = \frac{b - a}{n}$
- Height: $f(x_i^*)$ where $x_i^*$ is a sample point in the $i$-th subinterval
The Riemann Sum is:
$$S_n = \sum_{i=1}^{n} f(x_i^*) \cdot \Delta x$$
As we let $n \to \infty$, these rectangles become infinitely thin and the approximation becomes exact.