7.2 Definite Integrals and Properties of Integrals

In this lesson, you will learn about the definite integral and its relationship to the area under a curve. You will also learn to use properties of the definite integral.

Lesson Objectives

Upon completion of the lesson 7.2, you will be able to:

- Recognize that the areas of regions with irregular boundaries are limits of Riemann sums.
- Identify the definite integral of a function for a given limit of a Riemann sum of a function.
- State the properties of definite integrals.
  - The definite integral of a function over single point equals zero. .

- - Interchanging the limits of integration changes the sign of the integral.
  - Constant multiples can be factored out of the integrand of an integral.
  - The integral of a sum or difference is the respective sum or difference of integrals.

- Use the geometry of a graph to evaluate a given definite integral. (Use of the areas of Circles, rectangles, triangles, and trapezoids.)
- Find the value of a definite integral given the area of regions between the curve and the x-axis.

Lesson Content

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

1. YouTube video: The Definite Integral

1. YouTube video: Riemann Sum and the Definite Integral

1. The Integral (from University of Houston)

1. Note: The general Riemann sum does not require that the rectangles have uniform width, . Therefore, the Riemann sum in this video uses for the width of the ith rectangle. Furthermore, the general Riemann sum does not require that we use the left hand endpoint, right hand endpoint, or midpoint to determine the "height" of the rectangle. Therefore, the Riemann sum in this video uses

1. to represent the x-value in the ith interval that is used to find the "height" of the rectangle, .

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. The Definition of the Definite Integral [Paul's Online Math Notes]
2. Note: Read only until the end of "More Properties."

The Definite Integral [University of Minnesota]

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. Definition of the Definite Integral [Paul's Online Math Notes]
2. Note: Do problems # 1, 4 - 9.
3. Definite Integrals [University of New England] (from University of New England) Watch the video and do the practice problems. Check the answer and explanation for each problem.
4. The Limit Definition of a Definite Integral [UC Davis]
5. Note: Read about the definite integral as a limit then do problems #1, 2, 4, 9, 10, 12, 13.

Additional Resources

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

1. The Limit of a Riemann Sum - Applet [Wolfram]
2. Note: Explore the approximation of the definite integral of a function using rectangles and a Riemann sum. Try large values for the number of rectangles to simulate the concept of taking the limit as the number of rectangles increase and the width of each rectangle approaches zero. Be careful - if you select too large a value for the number of rectangles, the graph will shut down because it can't handle it. Don't go over 1000 rectangles.
3. YouTube videos: Exact Area Under a Curve - Find the limit of a Riemann Sum
4. Part 1:

1. Part 2:

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