The Definite Integral

Accumulating Under a Curve

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

It is defined as a limit of Riemann sums. Partition $[a, b]$ into $n$ thin subintervals of width $\Delta x$, each with a sample height $f(x_i)$ from the curve. The sum of rectangle areas $\sum f(x_i)\,\Delta x$ approximates the region, and as the slices get thinner ($n \to \infty$) it becomes exact: $$\int_a^b f(x)\, dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i)\,\Delta x.$$ (The integral sign is an elongated "S" for "sum.")

Two features to remember:

• Signed area. Where $f(x)$ is below the axis, its contribution is negative. The definite integral gives net signed area, so parts below the axis subtract. A symmetric curve can integrate to zero while enclosing real area.
• A number, not a function. Unlike the indefinite integral (a family of functions with $+C$), the definite integral is a single number, with no constant of integration.

Basic properties follow from the area picture: $\int_a^a f = 0$, reversing limits flips the sign ($\int_b^a f = -\int_a^b f$), and integrals split over adjacent intervals ($\int_a^c = \int_a^b + \int_b^c$). Numerically, Riemann sums and refinements (trapezoidal rule, Simpson's rule) are how computers evaluate integrals with no closed form.

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 it can be zero or negative even when real area is enclosed. And it 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.

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