The Definite Integral

Accumulating Under a Curve

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

It is defined as the limit of Riemann sums, partitioning $[a,b]$ into $n$ subintervals of width $\Delta x$:

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

Here, $\Delta x$ is subinterval width, $f(x_i)$ is sample height, and the integral sign is an elongated "S" for sum.

Signed Area and Properties

Common Pitfall: Treating definite integrals as purely geometric positive area. Parts below the axis subtract from the total.