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Integration and differentiation

I took advanced placement (AP) calculus in high school and it was a mistake.  For I nearly failed my first college physics course; so I took calculus again.  Now, as a professor of mechanical engineering, I see many students making the same mistake I made.  They seem to hold a view that the two fundamental strands of calculus are merely inverses of each other.

There are historical steps in the development of integration.  The "method of exhaustion," by Eudoxus, was the first attempt to determine integrals in his study of celestial motion.  This was followed by the use of mathematical induction by Ibn al-Haytham while studying optics.  Next was the "method of indivisibles" by Cavalieri in the 16th Century.

There are also historical steps in the development of differentiation.  Its conception as a tangent was familiar to Euclid.  The use of infinitesimals to study rates of change were studied in India  by Aryabhata.   Sharaf al-Dīn al-Tūsī in Persia studied the derivatives of polynomials.

Finally, Gregory, Barrow and Newton linked these two strands with the first fundamental theorem of the calculus.  From my own discussions with students, I have learned that many see the two strands as inverses of each other, often using the phrase "anti-differentiation" to describe integration.  Steeped in simplistic AP calculus courses that place undue emphasis on mechanistic learning, students rarely transcend that practical, although pedagogically necessary, understanding.

Consider deformation.  An elastic material is defined by a linear algebraic relationship between force and displacement.  A hyperelastic material is represented by an integral: it is the sum of all possible effects that contribute to material deformation. A hypoelastic material is represented by a derivative which enables one to separate deformation from rotation.  Integration and differentiation is used to create different material models.

Consider dynamics.  The derivative enables the study of a particle's trajectory.  Integration enables work/energy relationships that provide information on initial and final configurations.  Again, integration and differentiation are used purposefully: to model motion different ways.

We should dissuade some high school students from rushing into calculus classes until they have the maturity to appreciate calculus, almost as a philosophy.  AP courses should refrain from simply describing integration and differentiation as opposite processes, and demonstrate that they can be used to model the world different ways.

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