<p>This paper provides a new categorification of the Lebesgue integral with variable upper limits by using normed modules over finite-dimensional <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {k}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">k</mi> </math></EquationSource> </InlineEquation>-algebras <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\( \Lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation> and the category <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathscr {A}^p_{ \Lambda }\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="script">A</mi> <mi mathvariant="normal">Λ</mi> <mi>p</mi> </msubsup> </math></EquationSource> </InlineEquation> associated with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\( \Lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation>. The integration process is redefined through the introduction of an integral partially ordered set and an abstract integral with variable upper limits. Finally, we present two important applications: (1) the categorification of basic elementary functions, including (anti-)trigonometric and logarithmic functions, and (2) a new approach for characterizing the global dimensions of gentle algebras.</p>