Lesson Plans

Calculus: Graphical, Numerical, Algebraic ©1999

by Finney

Week 16

Chapter 4 Test: Applications of Derivatives
Chapter 5, Section 5.1: Estimating with Finite Sums; Section 5.2: Definite Integrals

College Board Objectives:

• Analysis of graphs. With the aid of technology, graphs of functions are often easy to produce. The emphasis is on the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function.
• Riemann sums.
  • Concept of a Riemann sum over equal subdivisions.
  • Computation of Riemann sums using left, right, and midpoint evaluation points.
• Interpretations and properties of definite integrals.
  • Definite integral as a limit of Riemann sums.
  • Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval:
    f ´(x)dx = f(b) – f(a)
  • Basic properties of definite integrals. (For example, additivity and linearity.)
• Applications of integrals. Appropriate integrals are used in a variety of applications to model physical, social, or economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the integral of a rate of change to give accumulated change or using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region, the volume of a solid with known cross-sections, the average value of a function, and the distance traveled by a particle along a line.
• Numerical approximations to definite integrals. Use of Riemann sums and the Trapezoidal Rule to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values.

Resources:

• Chapter 4 Test: Applications of Derivatives
• Chapter 5, Section 5.1: Estimating with Finite Sums; Section 5.2: Definite Integrals—pp. 246–266
• Teacher's Guide with Answers—pp. 43–47
• Advanced Placement* Correlations and Placement–Concepts Worksheets—none
• Assessment—pp. 23–26

Pacing Guide:

• 1st day—Test
• 2nd day—Section 5.1
• 3rd day—Section 5.1
• 4th day—Section 5.1
• 5th day—Section 5.2

Key Words:

Designing Lessons/Student Responses to Lesson:

• 5.1—The Quick Review exercises provide an excellent introduction to this lesson. Have students discuss how these problems are related to the area concept. Several methods of using rectangles to estimate the area under the graph of a nonnegative continuous function are introduced in this section. It is important that students graph the curve over the desired interval in order to visualize the area being sought. The built-in shade feature of many graphers can be helpful here. You may wish to introduce sigma notation in order to allow for a simple and effective way to express the sums used in finding RAM approximations. This notation will be used in the definition of the definite integral. Exploration 1 is important because it enhances the understanding of how estimates obtained using various RAM methods are related to an actual area under a curve. Students will learn that LRAM underestimates the area under the graph of an increasing function but overestimates the area under a decreasing function, while RRAM does the opposite. For small values of n, students should be able to calculate LRAM, MRAM, and RRAM by hand. For large values of n, students should use a RAM program. You may also wish to help students discover ways to use the list menu on a grapher in calculating various RAM estimates. As n increases, students will see the approximating sums converge to a limit, which, after all, is the whole idea of integral calculus. An understanding of the RAM method can lead to a much greater appreciation of the Fundamental Theorem. Example 4 provides a meaningful conclusion to this lesson.
• 5.2—A review of sigma notation is essential to the understanding of this lesson. You may wish to use the Quick Review exercises for this purpose. The formal definition of the definite integral as a limiting value of Riemann sums is presented in this section. To understand how the definition works, it is easiest to first assume that the function is nonnegative. In that case, the value of the definite integral is the exact area under the curve, and a Riemann sum is a rectangular approximation of that area. Students should understand that in the Riemann sum, f(c_k) and x_k represent the height and width of the kth rectangle, so the Riemann sum represents the sum of the areas of n rectangles. As the norm of the partition approaches 0, these approximating Riemann sums approach the exact value of the area. Emphasize that LRAM, MRAM, and RRAM are examples of Riemann sums. Students should then discuss the case when the function takes on negative values at some or all of the points in the interval of integration. The main idea to establish here is that the definite integral represents the signed or net area of the region between the graph of the function and the x-axis. Students may think the proof of Theorem 2 is unnecessarily tedious, since c(b – a) is clearly the area of the rectangles according to the simple geometry formula, A = bh. In some sense, what this proof shows is that the area, defined as an integral, is in fact the same as our usual notion of area. Note that, while integrals are commonly thought of as areas, there are many different kinds of quantities that can be represented as integrals. Be sure that students understand Exploration 1, which provides a conceptual understanding of the integral. Many graphers have a built-in numerical integrator. Make sure that each student can translate from a definite integral to the notation utilized by his or her grapher and vice versa. Although the NINT notation is very convenient, it is important to note that the Advanced Placement* test requires students to use integral notation, not calculator notation, for credit on free-response questions.

Critical Thinking Questions:

• 5.1—pp. 256–257, # 20, 22, 24–25
• 5.2—p. 268, # 47

Troubleshooting Tips/Error Traps:

• Some students may assume that the MRAM estimate will always be the average of the LRAM and RRAM estimates. Give an example (any quadratic function will do) to show that this is not the case.
• In writing definite integrals, students will often omit the dx.
• Some students will assume that values reported by NINT are always very nearly exact. NINT is likely to fail when a function is nearly zero except in a very small portion of an interval, as in e^–xdx or e^–x2 dx.