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I've lectured integration from the very basics, having covered the following;

  • What are integrals, and what do they represent?
  • Indefinite integrals as the opposite of derivatives.
  • Using the power rule for derivatives to provide the power rule for integration.
  • Integral as infinite sum.
  • Area under curve, and area between two curves.
  • Volume of solid of revolution.
  • Integration by substitution.
  • Integration by parts.
  • Integration by partial fraction expansion.
  • Improper integrals

All this is a total of 41 short (5-10 min) lectures (videos), accompanied by example problems etc.

I'm wondering what I should do next, such that it will be natural step forward. Any ideas?

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    $\begingroup$ This looks like a pretty standard Calculus sequence. Why not look at what's on the AB/BC AP tests? To re-phrase: Why would your progression differ from the standard ones in high school or college intro to Calc classes? If it wouldn't, this should be trivial (e.g., imitate James' text); if it would differ, then perhaps you can explain how in the body of your question. $\endgroup$
    – Benjamin Dickman
    Apr 9, 2014 at 9:51
  • $\begingroup$ I'm not decided on whether or not it would differ. So far, I've come up with the plan by myself. I tried looking at the syllabus for AP Calc, but it was very vague, and didn't really cover the sub-sections. $\endgroup$
    – Alec
    Apr 9, 2014 at 9:59
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    $\begingroup$ trigonometric substitution and/or hyperbolic substitution. The idea of implicit verses explicit substitution. $\endgroup$
    – James S. Cook
    Apr 9, 2014 at 14:01
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    $\begingroup$ Application in sciences, e.g. physics. $\endgroup$
    – Toscho
    Apr 10, 2014 at 21:52
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    $\begingroup$ Approximating sums by integrals. $\endgroup$
    – Toscho
    Apr 10, 2014 at 21:53

2 Answers 2

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I would move integral as infinite sum to the second topic, with crude numerical calculations, or by drawing a curve on graph paper and counting how many squares are covered.

I would add:

  • techniques for numerical integration**, e.g. trapezoidal rule and Simpson's rule.

  • integrating by computer, for both symbolic and numerical results.

  • applications in physics: distance from velocity, impulse from force, moments.

  • applications in biology or economics, e.g. population growth as the integral of population changes, or GDP growth as the integral of changes in GDP.

  • applications in statistics, e.g. normal probabilities as the area under the normal distribution.

I would remove

  • integration by partial fractions, which I find unimportant and distracting for a course at the level of "What are integrals?"

  • integration by parts, for the same reason

  • volumes of revolution, if pressed for time, because it fits better in the multivariable class

  • improper intergals, if pressed for time, because students are unlikely to appreciate the subtleties

That keeps the total lectures at roughly the same length, with a different emphasis.

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  • $\begingroup$ I would keep integration by parts, only because of how important it is for later math. A large part of my work as an analyst seems to be integration by parts... $\endgroup$
    – Steven Gubkin
    Jun 13, 2014 at 17:03
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As Michael E2 mentions, a next step could be numerical integration, introducing methods such as the rectangle rule, trapezoidal (trapezium) rule and Simpson's rule.

(I have made this answer Community Wiki.)

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