---
tag: 0N91N98EG5
kind: statement
role: theorem
name: Mean value theorem for integrals
msc: [26A42]
uses: [00KAFBR32E, 0WS1A8E2P9, 0MBWQKM8QR]
sources:
  - url: https://en.wikipedia.org/wiki/Mean_value_theorem
    locator: First mean value theorem for definite integrals
---

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be continuous.
Then there is $c \in [a, b]$ with
$$\int_a^b f(x) \, dx = f(c) \, (b - a).$$

## Remarks

Let $m$ and $M$ be the minimum and the maximum of $f$, from the [[extreme value theorem|0WS1A8E2P9]]. By [[monotonicity|0MBWQKM8QR]], the average $\frac{1}{b - a} \int_a^b f$ lies in $[m, M]$. The [[intermediate value theorem|00KAFBR32E]] gives $c$.
The number $\frac{1}{b - a} \int_a^b f$ is the average value of $f$ on $[a, b]$.
Mathlib has a form for averages, `exists_eq_interval_average`.

## Explanations

### L1 human

A continuous function takes its average value somewhere.
The area under the graph equals the area of a rectangle whose height is a value of $f$.