Compact interval/Riemann integrable/Definite integral/Definition

Definite integral

Let ${}I=[a,b]$ denote a compact interval. For a Riemann-integrable function

$f\colon I=[a,b]\longrightarrow \mathbb {R} ,t\longmapsto f(t),$

we call the upper integral of ${}f$ (which by definition coincides with the lower integral) the definite integral of ${}f$ over ${}I$. It is denoted by

$\int _{a}^{b}f(t)\,dt{\text{ or by }}\int _{I}^{}f(t)\,dt.$