Definition:Multiple Integral

Definition

Let $n \in \N$ be a natural number.

Let $R \subseteq \R^n$ be a closed rectangle in $\R^n$.

Let $f : R \to \R$ be a bounded real-valued function on $R$.

Darboux Integral

Suppose that:

$\ds \underline{\int_R} \map f x \rd x = \overline{\int_R} \map f x \rd x$

where $\ds \underline{\int_R}$ and $\ds \overline{\int_R}$ denote the lower Darboux integral and upper Darboux integral, respectively.

Then the multiple Darboux integral of $f$ over $R$ is defined and denoted as:

$\ds \int_R \map f x \rd x = \underline{\int_R} \map f x \rd x = \overline{\int_R} \map f x \rd x$

and $f$ is (properly) multiple integrable over $R$ in the sense of Darboux.

Also known as

A multiple integral is also known as an iterated integral.

Also see

• Results about multiple integrals can be found here.