User:Keith.U/Sandbox/SubSandbox 2/SubSubSandbox 2

Definition
Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval.

Let $f: \left[{a \,.\,.\, b}\right] \to \R$ be a real function. Let $f$ be bounded on $\left[{a \,.\,.\, b}\right]$.

Let $f$ be Riemann integrable over $\left[{a \,.\,.\, b}\right]$.

The Darboux integral of $f$ over $\left[{a \,.\,.\, b}\right]$ is denoted
 * $\displaystyle \int_a^b f \left({x}\right) \, \mathrm d x$

and is defined as:
 * $\displaystyle \int_a^b f \left({x}\right) \, \mathrm d x = \underline{\int_a^b} f \left({x}\right) \ \mathrm d x = \overline{\int_a^b} f \left({x}\right) \, \mathrm d x$

where $\displaystyle \underline{\int_a^b}$ and $\displaystyle \overline{\int_a^b}$ denote the lower integral and upper integral, respectively.