A traditional approach to Calculus starts with the analytic foundations. This time tested method introduces the critical concepts of limits, continuity, and the completeness of the Real numbers as the foundation for the development of the derivative and integral. This development culminates in the proofs of the fundamental theorems of Calculus linking the derivative and the integral.
With the fundamental theorem, each known derivative can be reinterpreted as an integral. Various techniques of integration are presented and the student is invited to practice these in the exercises. The course then moves on. Perhaps to polynomial approximation, or Calculus in higher dimensions.
Integration remains a challenge.
The experiment is to bring integration early in the development. This is done by deferring the analytic foundations, and presenting differentiation and integration, more precisely antidifferentiation, in formal terms. This allows the reader to get to integration quickly and to focus on developing familiarity with the techniques of integration.
The analytic foundations are hinted at as the motivation for the the formal derivative of a function is based on linear approximations to the function. Likewise the motivation for the antiderivative of a function is based on the area under that function.
With the formal machiney developed and understood, the analytic definitions can be presented. The formal rules are proved, and with that all the techniques are on solid analytical ground.
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