Definition:Upper Darboux Integral/Rectangle

Definition

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

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

The upper Darboux integral of $f$ over $R$ is defined as:

$\ds \overline{\int_R} \map f x \rd x = \inf_P \map U P$

where:

$P$ ranges over all finite subdivisions of $R$.

$\map U P$ denotes the upper Darboux sum of $f$ on $R$ with respect to $P$.

Sources

• 1965: Michael Spivak: Calculus on Manifolds: $3$ Integration