# Teaching integration (online). Logical progression in subject?

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?

• 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. – Benjamin Dickman Apr 9 '14 at 9:51
• 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. – Alec Apr 9 '14 at 9:59
• trigonometric substitution and/or hyperbolic substitution. The idea of implicit verses explicit substitution. – James S. Cook Apr 9 '14 at 14:01
• Application in sciences, e.g. physics. – Toscho Apr 10 '14 at 21:52
• Approximating sums by integrals. – Toscho Apr 10 '14 at 21:53

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.

• 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.

• 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... – Steven Gubkin Jun 13 '14 at 17:03

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.