All units are integral operators

I've been using dimensional analysis since high school, yet I only just grokked that all units (metres, seconds, amperes) are in fact integrations over functions. When we write newton-metres we actually mean the operator (\int \int dN . dm), and kg/s means (\int dkg) (d/ds).

The differentiation example is commonly given in calculus (rates of change, velocity and acceleration) but the idea that m2 is actually an implicit double integral is only hinted at by most textbooks.

The simple approach of multiplying numbers with units (say speed times elapsed time) is a shorthand for performing an integration on a function with no dependance.