Calculating Areas Using Integrals
Big Idea: Integration allows us to calculate the exact area of regions bounded by curves — something that basic geometry formulas cannot handle. This is one of the most powerful applications of calculus.
1. The Fundamental Concept
Recall that the definite integral of a function $f(x)$ from $a$ to $b$ is defined as:
$$ \int_a^b f(x)\, dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x $$
Geometrically, when $f(x) \geq 0$ on $[a, b]$, this integral gives the area under the curve $y = f(x)$, above the $x$-axis, between $x = a$ and $x = b$.