$$ \newcommand{\RR}{\mathbb{R}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\CC}{\mathbb{C}} \newcommand{\NN}{\mathbb{N}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\FF}{\mathbb{F}} \renewcommand{\epsilon}{\varepsilon} % ALTERNATE VERSIONS % \newcommand{\uppersum}[1]{{\textstyle\sum^+_{#1}}} % \newcommand{\lowersum}[1]{{\textstyle\sum^-_{#1}}} % \newcommand{\upperint}[1]{{\textstyle\smallint^+_{#1}}} % \newcommand{\lowerint}[1]{{\textstyle\smallint^-_{#1}}} % \newcommand{\rsum}[1]{{\textstyle\sum_{#1}}} \newcommand{\uppersum}[1]{U_{#1}} \newcommand{\lowersum}[1]{L_{#1}} \newcommand{\upperint}[1]{U_{#1}} \newcommand{\lowerint}[1]{L_{#1}} \newcommand{\rsum}[1]{{\textstyle\sum_{#1}}} % extra auxiliary and additional topic/proof \newcommand{\extopic}{\bigstar} \newcommand{\auxtopic}{\blacklozenge} \newcommand{\additional}{\oplus} \newcommand{\partitions}[1]{\mathcal{P}_{#1}} \newcommand{\sampleset}[1]{\mathcal{S}_{#1}} \newcommand{\erf}{\operatorname{erf}} $$

Integrals

• In 28  Axioms we discuss the area problem and give a set of axioms to characterize the operation of integration.
• In 29  Construction we give a particular limiting construction (the Darboux Integral) and prove it satisfies the axioms
• In 30  \bigstar Examples we integrate polynomials and elementary functions directly from the definition
• In 31  Integrable Functions we investigate further properties of the Darboux Integral beyond the axioms; showing that it is a linear operation and all continuous functions are integrable