Definition:Multiple Integral/Darboux

Definition

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Let $R$ be a closed rectangle on $\R^n$

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

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.

Sources

• 1965: Michael Spivak: Calculus on Manifolds