Every text that I read starts by defining differentiation then integration... but does anyone know if there is one that goes the other way? Also is there any harm in taking this approach.... to me, the current approach is only done because it has always been this way. I say this because I believe that this approach would allow a neater flow of logic in teaching calculus.
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Richard Courant's Differential and Integral Calculus discusses the integral before differentiation.
as Mike noted: Apostol, Calculus (1967) begins with integration.
But then, later, derivatives are defined in the usual way, not somehow obtained as anti-integral.