---
tag: F961Q88FHK
kind: concept
name: Darboux integral
msc: [26A42]
sources:
  - url: https://en.wikipedia.org/wiki/Darboux_integral
same_as:
  wikidata: Q2294564
  wikipedia: Darboux integral
---

## Summary

The Darboux integral defines the integral of a bounded function on a closed interval by squeezing it between [[upper and lower Darboux sums|0QRQVH0CRW]].
A function is Darboux integrable when its [[upper and lower Darboux integrals|09FQT057T7]] agree, and the integral is the common value.
This gives exactly the same integrable functions and the same values as the [[Riemann integral|0QM4TXT3DR]], by [[a classical theorem|021XJAQVDC]].
It is often the most convenient way to prove that a given function is integrable.

## Explanations

### L1 human

Cut the interval into pieces and draw on each piece a rectangle as high as the largest value of the function there, and another as high as the smallest value.
The tall rectangles cover the area under the graph from above, and the short ones fill it from below.
If refining the cuts can bring the two totals as close as you like, the area is the number caught between them.
For $f(x) = x$ on $[0, 1]$ with $n$ equal pieces, the two totals are $\frac{n+1}{2n}$ and $\frac{n-1}{2n}$, and both tend to $\frac12$.