Before I write about integration, I’d like to say a few words about the order of topics. Since I assume most of you are AP Calculus teachers, the list of concepts and topics goes something like this:
- Area of a region between the graph and the x-axis, which leads to
- Riemann sums, which leads to
- The definition of the definitive integral.
- Numerical integration – left-, right-, upper-, lower-, midpoint sums. The Trapezoidal Rule and Trapezoidal approximations. Integration with technology.
- The Fundamental Theorem of Calculus (FTC)
- Applications of Integration
- Introduction to differential equations.
Your preferred order of topics may be slightly different.
What is missing from this list is antidifferentiation or techniques of integration. There are several places where teachers have placed it with successful results.
- Many books and therefore many teachers place the topic of antidifferentiation at the very end of the derivative work or the very beginning of the integration. This has the advantage of having your students know how to do at least simple antidifferentiation when they need it right after the FTC. On the other hand, at this point students may not see the need for it and see it as the “game” of reverse differentiation.
- A more natural place to tackle it is immediately after the FTC. The advantage here is that now students have a reason to want to antidifferentiate so they can evaluate definite integrals.
- Other teachers change the order above slightly and teach applications before the FTC. They set up Riemann sums and definite integrals for the various applications and then have the student evaluate the integrals on their calculators.
In the usual order listed above each application problem has two parts: the first is to set up the integral and then the antidifferentiation to evaluate it. Often the antidifferentiation is the longer part. By using technology to do the evaluation, students need only concentrate on setting up the correct definite integral and quickly do the evaluation on their calculator.
Once the students are good at applications they then go on to the FTC and antidifferentiation as separate new topics. While learning antidifferentiation techniques teachers can assign one or two applications each night so students get more practice (spiraling).
I do not advocate one or the other of these approaches. They all have been tried and they all work. I am just pointing out the different ways so you will know that there is a choice. Pick the order you are comfortable with or pick a new order if you want to try something different.