Defining and Computing Definite Integrals



Defining and Computing Definite Integrals

Overview: In this lesson, students will learn what a Riemann sum is and be given a step-by-step procedure of how to formulate them. They will also learn how to calculate both upper and lower Riemann sums. They will examine Riemann sums in depth through geometer’s sketchpad. A formal definition of a definite integral will be discussed and students will learn and use integral notation – integrand, limits of integration, variable of integration, and what it means for an integral to be Riemann integrable. The students will also discover the properties of definite integrals and explore how to use graphs to calculate definite integrals over intervals. Finally, they will learn how to calculate the worst possible error of an approximation integral.

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

Time: 3-50 minute class periods

Purpose: The purpose of this lesson is to define definite integrals using Riemann sums. By doing this, students will truly understand how integrals work rather than just learning a formula. They will be able to make very good approximations using Riemann sums and compare their approximations to the real answers in order to calculate their error. They will also learn many properties of the definite integral which will help them to perform their integrations faster.

Prerequisite Knowledge:

Students should:

- Understand the summation notation

- Understand the basics of Geometer’s Sketchpad

Objectives:

1. Students will learn how integrals work through Riemann Sums, instead of just memorizing a formula.

2. Students will be able to calculate Upper and Lower Riemann sums and the largest possible error in these approximations.

3. Students will be able to understand and use integral notation-integrand, limits of integration, and variable of integration.

4. Students will understand and be able to use the properties of definite integrals

5. Students will learn how to use graphs to calculate definite integrals over intervals.

Standards:

1. Communications: Students will be asked to communicate what they learn through homework and through the Geometer’s Sketchpad activity.

2. Problem-Solving: Students will use their problem solving skills in order to calculate upper and lower Riemann Sums and to approximate area using different geometric shapes.

3. Technology: Students will use Geometer’s Sketchpad to explore Riemann Sums.

Resources/Materials Needed:

1. Computer Lab with Geometer’s Sketchpad

2. Handouts for Geometer’s Sketchpad Lab

3. Calculus Book

4. Paper/Pencil

Activities and Procedures:

1. Introduce Riemann sums. Draw a function on the board and draw in the rectangles to give the students a graphical representation of what a Riemann Sum is.

2. Go through the steps of producing a Riemann Sum.

Step 1: For some positive integer n, subdivide the interval [a,b] into n subintervals of equal length. This forms what is called regular partitions of [a,b] of size n. Each subinterval length, [pic].

If we label our starting point as a = a0. The partition points would be:

a0 = a, a1 = a +[pic], a2 = a +2 [pic], a3 = a + 3 [pic], …, an = b = a + n [pic]

Step 2: Choose one point xi (i = 1, 2,…,n) from each of the n subintervals. Each endpoint may be chosen from anywhere in the subinterval, including the endpoints. Common choices are the left or right endpoint, or the midpoint of each subinterval.

Step 3: Calculate f(xi) for each xi chosen, multiply each function output by [pic], and sum the results:

[pic]

This is the value of the Riemann sum for f over [a, b] for this partition and choice of points.

3. Have the students calculate the Riemann sum of f(x) = x2 over the interval [0,3] with a regular partition of n = 4 subintervals and with chosen points x1 = 0.5, x2 = 1.0, x3 = 2.0, x4 = 2.7.

4. Explain how summation notation can be used for donating Riemann sums.

5. Introduce Upper and Lower Riemann Sums. A lower Riemann sum corresponds to the sum of the areas of the inscribed rectangle. An upper Riemann sum corresponds to the sum of the areas of the circumscribed rectangle. If a continuous function is monotonically increasing or decreasing, the minimum and maximum output values will always occur at the endpoints of each subinterval.

6. Have each group find the upper and lower Riemann sums for f(x) = x2 over the interval [0, 3] with a regular partition of n = 4 subintervals.

Day one Homework: Read section 6.2 and complete the attached worksheet #1

Day 2:

7. Go over Worksheet from previous Day

8. Riemann Sum Activity Day: Go to computer lab and let students explore Riemann sums through Geometer’s Sketchpad. The Lab worksheet is due at the beginning of class the following day.

Day 3:

9. Go over Lab Activity from previous day.

10. Give the formal definition of a definite integral. Let f be a function defined on a closed interval [a, b]. If the Riemann sums for f over [a, b] converge on a single limiting value A as the partition gets finer and finer then we write [pic] and we say that f is Riemann integrable over the interval [a, b]. Here f(x) is called the integrand, and a and b are the limits of integration, and dx denotes the variable of integration.

11. Have each student compute the following definite integral. Approximate the upper and lower Riemann sums for [pic] using a regular partition of size n = 5. Once they have these two values, explain how to average the upper and lower approximations. Sometimes this average will be a better approximation, sometimes it won’t be.

12. Discuss how to find the largest possible error.

Largest possible error =[pic]. Calculate this error from the previous example.

13. Properties of Definite Integrals:

a. Additivity over Intervals: If a ................
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