Definite Integral
Calculus & Advanced MathA 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. It gives a single number as the answer.
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 $3$ is $9$ square units.
Key Insight
Unlike the indefinite integral (which gives a family of functions), the definite integral gives a single number representing an actual area.
Definition
The definite integral $\int_a^b f(x)\, dx$ equals $F(b) - F(a)$ 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."
Example
$\int_0^\pi \sin(x)\, dx = [-\cos(x)]_0^\pi = -\cos(\pi) + \cos(0) = 1 + 1 = 2$.
Key Insight
The Fundamental Theorem converts the definite integral from a limit of sums into a simple subtraction, making exact calculation tractable.
Definition
The Riemann integral of $f$ on $[a,b]$ is the limit of Riemann sums as the partition norm approaches $0$, provided the limit exists and is independent of partition choice. The Lebesgue integral extends this to a broader class of functions using measure theory, agreeing with the Riemann integral for all Riemann-integrable functions.
Example
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
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.