Definite Integral
A definite integral computes the exact net area between a function and the x-axis over a specific interval [a, b].
Formula
\int_a^b f(x)\, dx = F(b) - F(a)
Definition
A definite integral finds the exact net area between a curve and the x-axis over a specific stretch from $x = a$ to $x = b$, giving a single number as the answer rather than a family of functions. It equals $\int_a^b f(x)\, dx = F(b) - F(a)$ where $F$ is any antiderivative of $f$ (by the Fundamental Theorem of Calculus); areas below the $x$-axis count as negative, making it the "net signed area." More rigorously, it is the limit of Riemann sums as the partition norm approaches $0$, provided the limit exists independent of partition choice; the Lebesgue integral extends this to a broader class of functions using measure theory.
Example
$\int_0^3 2x\, dx = [x^2]_0^3 = 9 - 0 = 9$, the area of the triangular region under $y = 2x$ from $0$ to $3$. Signed area matters: $\int_0^\pi \sin(x)\, dx = [-\cos(x)]_0^\pi = 1 + 1 = 2$. The Dirichlet function ($1$ on rationals, $0$ on irrationals) is not Riemann integrable but is Lebesgue integrable, with integral $= 0$ on $[0,1]$ since the rationals have measure zero.
Key Insight
Unlike the indefinite integral, which gives a family of functions, the definite integral gives a single number representing an actual area, and the Fundamental Theorem converts it from a limit of sums into a simple subtraction, making exact calculation tractable. Switching to Lebesgue integration resolves convergence theorems (Dominated Convergence, Monotone Convergence) that Riemann integration cannot handle, forming the foundation of modern probability and functional analysis.