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.