Topics 10.1, 10.2, 10.5
Identify the conditions that result in an improper integral.
Use limits to evaluate improper integrals or determine that the integral diverges.
Quick Lesson Plan
This is an exciting lesson for your students because their ideas about infinity will be pushed to their limits (pun intended!). As students have never considered the notion of an integral having infinity as a limit, we set the stage by posing the question:: can an infinite region actually have a finite area? In parts a-d of question 1, students will estimate the area of bounded areas. In part e, they will transition from finite areas to a region that is unbounded. In question 2, students find the general antiderivative and consider various values of the upper limit of integration, b. Students should note that the areas continue to increase, albeit slowly, as b increases without bound. Though they can’t directly substitute ∞ into the equation, the table will help them recognize they are looking for very, very, very large values of b. Furthermore, the use of the word “approaching” in question 3 will alert students that limits may be involved here. At the first stop sign, pause the class for a whole-class debrief. Encourage students to share their findings and whether they were surprised by the results and why. You can use the Monitoring Questions below to help you plan your debrief. In the second part of the lesson, the resulting area does have a finite result despite the fact that the region is again unbounded. Helping students to make the connection between the two problems will benefit them as they consolidate what they have learned.
What is happening to the areas as b gets bigger? Why does this happen?
Do you think the areas will ever stop growing? Why or why not?
What do you know about the graph of ln x and why is that helpful here?
Why do you think the area does/doesn’t exist?
How can we approach the area at infinity?
What made the second area have a value while the first did not?
Can you summarize the process you used here?
Once students realize that unbounded regions can have a finite area, discussion of the process to evaluate improper integrals can occur. Some important ideas to emphasize:
The “bad” value may be infinity or may be a value that is undefined due to a vertical asymptote.
The “bad” value may be within the integral being integrated.
The strategy is to use a limit to approach the “bad” value since we cannot evaluate a function there.
Communication is a key element of improper integration. Students should not try to take shortcuts to get to an answer without showing each step of the process, especially the correct limit notation attached to the integral. This limit notation needs to be present during each step of the integration process until the limit itself is actually calculated. A second potential pitfall is the use of infinity as a number when evaluating an expression such as in the second example. This type of misuse of infinity may be penalized on the AP exam due to poor communication.