Definite integral

Definition

Let $I$ be an interval, let $f \colon I \to \mathbb{R}$, and let $a, b \in I$. Assume that $f$ is Riemann integrable on the closed interval between $a$ and $b$. The definite integral of $f$ from $a$ to $b$ is the number $\int_a^b f(x) \, dx$ defined as follows. If $a < b$, it is the Riemann integral of $f$ over $[a, b]$. If $a = b$, it is $0$. If $a > b$, it is $-\int_b^a f(x) \, dx$.

Remarks

With this sign convention, $\int_a^b f + \int_b^c f = \int_a^c f$ holds for all $a, b, c$ in $I$. The numbers $a$ and $b$ are the limits of integration, and $f$ is the integrand. Mathlib writes the definite integral as ∫ x in a..b, f x, through intervalIntegral.

Explanations

L1 human

The definite integral of $f$ from $a$ to $b$ is a number. It measures signed area under the graph between $a$ and $b$. The order of the limits matters: swapping them changes the sign.