6
$\begingroup$

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?

$\endgroup$
  • 5
    $\begingroup$ 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 '14 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 '14 at 9:59
  • 3
    $\begingroup$ trigonometric substitution and/or hyperbolic substitution. The idea of implicit verses explicit substitution. $\endgroup$ – James S. Cook Apr 9 '14 at 14:01
  • 3
    $\begingroup$ Application in sciences, e.g. physics. $\endgroup$ – Toscho Apr 10 '14 at 21:52
  • 2
    $\begingroup$ Approximating sums by integrals. $\endgroup$ – Toscho Apr 10 '14 at 21:53
3
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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 '14 at 17:03
2
$\begingroup$

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.)

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.