---
tag: 0P4WX7Y502
kind: definition
name: Iterated integral
msc: [26B15]
defines: 0WZCHE1ZF9
uses: [0JPCHTXWG0]
sources:
  - url: https://en.wikipedia.org/wiki/Iterated_integral
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Iterated_integral
    locator: Versus multiple integrals
---

## Definition

Let $R = [a, b] \times [c, d]$ and let $f \colon R \to \mathbb{R}$.
Suppose that, for each $x \in [a, b]$, the function $y \mapsto f(x, y)$ is integrable on $[c, d]$, and that $F(x) = \int_c^d f(x, y) \, dy$ is integrable on $[a, b]$.
Then the iterated integral of $f$, first in $y$ and then in $x$, is
$$\int_a^b \int_c^d f(x, y) \, dy \, dx = \int_a^b \left( \int_c^d f(x, y) \, dy \right) dx = \int_a^b F(x) \, dx.$$
The iterated integral in the other order, $\int_c^d \int_a^b f(x, y) \, dx \, dy$, is defined in the same way.

## Remarks

Iterated integrals in three or more variables are defined by repeating this step.
The inner bounds may depend on the outer variable, as in $\int_0^1 \int_0^x f(x, y) \, dy \, dx$. This integrates over the triangle $0 \le y \le x \le 1$.
An iterated integral is different in principle from the [[double integral|0NKBMFC18R]], but [[Fubini's theorem|0R0T1VWH5X]] says that they are equal under suitable hypotheses.
The two orders of integration can give [[different values|0GJZE2E4RX]] when the hypotheses fail.

## Explanations

### L1 human

To compute an iterated integral, work from the inside out.
First integrate in $y$ as if $x$ were a constant; the result is a function of $x$ only. Then integrate this function in $x$.