Activities and Procedures:



Improper Integrals

Overview: In this lesson, student will continue to use their prior knowledge of limits in order to investigate how to solve improper integrals. They will learn the different types of improper integrals and how to solve each of them. They will also investigate and find guidelines for dealing with integrals involving (1/xp).

Grade Level/Subject: This lesson is for 12th graders in AP Calculus.

Purpose: This lesson will enable students to deal with integrals that span over vertical asymptotes or integrals whose limits of integration are infinite. It will expand the students’ ability to compute infinite areas under curves.

Prerequisite Knowledge:

Students should:

- already know and understand the concept of limits

- Have a solid understanding of integration and the methods of substitution and integration by parts.

Objectives:

1. Students will be able to solve any improper integral and answer the question of whether the integral converges or diverges.

2. The students will have their knowledge of integrals refreshed for the upcoming AP exam.

Standards:

1. Connections: Students will connect their past knowledge of limits and integration techniques to solve improper integrals

2. Problem-solving: Students will be asked to form their own rules about the convergence or divergence of integrals containing (1/xp).

3. Communication: Students will complete a group activity at the end of the lesson in which they will have to communicate with their fellow students in order to solve a problem.

Resources/Materials Needed:

1. Dry-Erase Board/Dry-Erase markers of varying colors

2. Calculus Book

3. Worksheet

Activities and Procedures:

1. [pic]So far we have dealt with integrals of the form [pic] where it is assumed that a and b are finite numbers and the function f(x) behaves nicely over the interval. However, this isn’t always the case. Improper integrals occur when:

• the function f(x) blows up (goes to [pic]) at one of the endpoints

• one of the end points a and/or b is infinite

• a combination of both of the above

2. Start with functions that are defined over unbounded intervals. These types of improper integrals are called horizontal type improper integrals or vertical type improper integrals. These can be thought of as the limiting value of a proper definite integral as one endpoint approaches [pic]or [pic]. In other words,

[pic]

|[pic] |Fig. 2.2 |

| |[pic] |

Or [pic]

|[pic] |Fig. 2.3 |

| |[pic] |

If a limiting value exists, then the improper integral converges. Otherwise, it diverges.

3. Examples:

a. [pic] = [pic] = [pic]

Therefore, this limit diverges.

b. [pic] = [pic]

Here we have to use integration by parts: Does anyone remember how to do this?

Let’s set u=x dv=e-xdx

du=dx v=-e-x

According to our integration by parts formula, we now have:

[pic]= [pic]( This limit converges.

c. [pic]( This limit converges.

d. [pic]( this limit converges.

4. There are also vertical type improper integrals, which are very similar to the horizontal type improper integrals. The difference is that vertical type integrals are over intervals that contain vertical asymptotes.

They look like this:  

|[pic] |Fig. 3.1 |

| |[pic] |

Or

|[pic] |Fig. 3.4  |

| |[pic] |

Example:

a. [pic]

Looks like we will need to use integration by substitution: u=(x-1)

du = dx

= [pic]

5. What happens if both limits of integration are infinite? We can simply write this as a sum of the two integrals we have just been talking about.

[pic]

c can be anything we choose, typically we put it at a place where f(x) has a line of symmetry or if it does not have symmetry, we typically choose c=0.

Example:

a. [pic]= [pic] +[pic] = [pic][pic] + [pic][pic]

Looks like we need to use Substitution to solve this one: u = ex

du=exdx

= [pic] +[pic] = [pic][pic] + [pic][pic] = tan-1(u)][pic] + tan-1(u)][pic] = [pic]

Class Activity:

At each of your tables, formulate a hypothesis regarding the convergence or divergence of [pic] for p < 1, p = 1, and p > 1.

Answer: Converges for p > 1, Diverges for p < 1 and p = 1.

Evaluation: Complete attached worksheet

[pic]

Name:

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. [pic]

10. [pic]

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