8.2 The Area Between Two Curves

In this lesson, you will learn how to find the area between two curves.

Lesson Objectives

Upon completion of the lesson 8.2, you will be able to use integration to calculate the area between two curves.

Lesson Content

View all of the following instructional videos. These will help you master the objectives for this module.

1. Areas Between Curves [University of Houston]

   1. YouTube video: Finding Areas Between Curves

• 1. YouTube videos: Area Between Curves - integrating with respect to y

  2. Part 1:

• 1. Part 2:

Lesson Readings

The following required readings cover the content for this module. As you go through each reading, pay close attention to the content that will help you learn the objectives for this module.

• 1. Area between curves (from Paul's Online Math Notes)

  2. Note: For example 2, read about how the integral is set up to calculate the area desired but since we have not learned how to differentiate or integrate the natural exponential function (y=ex), please skip the actual evaluation of the integral.

1. Area and Volume Tutorial

   1. Note: Read only the section titled Areas Between Two Curves. Also view the 3 examples at the end of that section.

Lesson Practice Exercises/Activities

Make your way through each of the practice exercises. This is where you will take what you have learned from the lesson content and lesson readings and apply it by solving practice problems.

• 1. Area between two curves [University of Tennessee] (problems with solutions)

  2. Note: Skip the 7th and 9th problem.

  3. Some Area Problems [University of Houston] (answers and solutions provided)

  4. More on Area

Additional Resources

Below are additional resources that help reinforce the content for this module.

• 1. Area Between Two Curves [Eastern Iowa Community Colleges]

  2. Note 1: Scroll down the page and read only the section titled "Determining the Area Between Two or More Curves."

• 1. Note 2: For example 6, recall the trigonometric identity cos 2x=1 - 2sin2x. If you solve for sin2x, you obtain . We use this identity to replace sin2 x with an expression that is more easily integrated.

  2. YouTube videos: Area of a Region Between Two Curves - Calculus

  3. Part 7.1a1:

• 1. Part 7.1a2: