The Definite Integral and Area

Accumulating Under a Curve

The definite integral measures the accumulated area between the curve $y = f(x)$ and the $x$-axis from $x = a$ to $x = b$. While derivatives measure instantaneous rates, integrals measure totals, like total distance traveled from a velocity curve.

Formally, it is the limit of Riemann sums:

$$\int_a^b f(x)\, dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i)\,\Delta x$$

Here, $\Delta x$ is the width of $n$ rectangles, $f(x_i)$ is their height, and $\int$ is an elongated "S" for sum.

Signed Area and Pitfalls

The definite integral calculates signed area. Where $f(x)$ sits below the $x$-axis, its contribution is negative. A symmetric wave above and below the axis can integrate to zero, despite enclosing real physical area.

Common pitfall: Confusing definite and indefinite integrals. If you need total geometric area of a wave crossing the axis, split the integral at the roots and take absolute values: $\int |f(x)|\, dx$.