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.
Nathan's Home page
I am a modern day Renaissance man with interests ranging from mathematics, through computer science and the physical sciences through to music and art, how we live in cities, and teaching problem solving.
I'm currently living in Seattle, USA.
Things people email me about regularly.
Water level sensor
Cross stitch pattern generator