The Definite Integral and Area

Accumulating Under a Curve

The definite integral $\int_a^b f(x)\, dx$ is the second face of integration: it measures the accumulated area between the curve $y = f(x)$ and the $x$-axis, from $x = a$ to $x = b$. Where the derivative was about instantaneous rates, the definite integral is about totals — the total distance from a velocity, the total accumulated from a rate.

The definition is a limit of Riemann sums: slice the interval $[a, b]$ into $n$ thin rectangles of width $\Delta x$, each with height $f(x_i)$ given by the curve. The sum of rectangle areas $\sum f(x_i)\,\Delta x$ approximates the region, and as the slices get thinner ($n \to \infty$) the approximation becomes exact: $$\int_a^b f(x)\, dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i)\,\Delta x.$$ This is why the integral sign $\int$ is an elongated "S" for "sum."

Two important features:

• Signed area. Where $f(x)$ is below the axis (negative), its contribution is negative. The definite integral gives net signed area, so regions below the axis subtract.
• A number, not a function. Unlike the indefinite integral (a family of functions with $+C$), the definite integral is a single number — there is no constant of integration.

Numerically, Riemann sums (and refinements like the trapezoidal rule and Simpson's rule) are exactly how computers evaluate integrals that have no closed form — a cornerstone of scientific computing.

Common pitfall: treating the definite integral as always giving the geometric (positive) area, and confusing it with the indefinite integral. The definite integral is signed — area below the axis counts as negative, so a symmetric curve can integrate to zero even though it encloses real area. And a definite integral is a single number (no $+C$), unlike the indefinite integral, which is a family of functions.

The area under a curve approximated by rectangles that grow thinner and more numerous, the accent shaded region converging to the exact integral, with a below-axis lobe shaded to subtract.

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