---
tag: 0QM4TXT3DR
kind: definition
name: Riemann integral
msc: [26A42]
defines: 0W29TBTY12
uses: [0N1JQXZXHC]
sources:
  - url: https://en.wikipedia.org/wiki/Riemann_integral
    locator: Riemann integral
---

## Definition

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be [[Riemann integrable|0N1JQXZXHC]].
The Riemann integral of $f$ over $[a, b]$ is the unique number $I$ in the definition of Riemann integrability.
It is written
$$\int_a^b f(x) \, dx.$$

## Remarks

For $f \ge 0$, the integral equals the area under the graph of $f$ and above the $x$-axis.
By convention, $\int_a^a f = 0$ and $\int_b^a f = -\int_a^b f$.
The [[fundamental theorem of calculus|0NHGY0N0JB]] computes the integral from an antiderivative.
The integral is named after Bernhard Riemann.

## Explanations

### L1 human

The Riemann integral is the limit of sums of areas of thin rectangles under the graph.
It gives the signed area between the graph and the $x$-axis.