Iterated integral

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, but Fubini's theorem says that they are equal under suitable hypotheses. The two orders of integration can give different values 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 $x$.