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Calculus: Graphical, Numerical, Algebraic ©2003
Ross L. Finney, Franklin D. Demana, Bert K. Waits, Daniel Kennedy
Week 17
Chapter 5, Section 5.2: Definite Integrals; Section 5.3: Definite Integrals and Antiderivatives
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.
- Fundamental Theorem of Calculus.
- Use of the Fundamental Theorem to evaluate definite integrals.
- Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined.
- 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 5, Section 5.2: Definite Integrals; Section 5.3: Definite Integrals and Antiderivatives, pp. 258–276
- Teacher's Guide with Answers—pp. 45–49
- Advanced Placement* Correlations and Placement–Concepts Worksheets—pp. 46–48
- Assessment—none
Pacing Guide:
- 1st day—Section 5.2
- 2nd day—Section 5.2
- 3rd day—Section 5.2
- 4th day—Section 5.3
- 5th day—Section 5.3
Key Words:
- average value of a function
- Mean Value Theorem for Integrals
Designing Lessons/Student Responses to Lesson:
5.3—A review of Example 5 in Section 5.2 can be used to open the lesson with a discussion of the Additivity rule for definite integrals. This rule, along with the other algebra rules of the definite integral are given without detailed proofs. Students should discuss their intuitive understanding of these rules. Note that the Max-Min Inequality in Table 5.3 can be derived as a consequence of the Domination rule: Let g(x) = min f. Then f(x) ≥ g(x) on [a, b], so f(x)dx ≥ g(x)ds = min f • (b – a). Similarly, f(x)dx ≤ max f • (b – a). The lesson concludes with a discussion which foreshadows the Fundamental Theorem of Calculus. Exploration 2 is a critically important visual proof of the Fundamental Theorem. This Exploration will help students achieve one of the goals of the course — to understand the relationship between derivatives and integrals. The historical perspective is intended to give students a sense of discovery as they work through Exploration 2. The function F(x) = 
f(t)dt is sometimes called the accumulation function of f.
Critical Thinking Questions:
5.3—p. 276, # 39–41, 44
Have students write a paragraph describing any difficulties they encountered while evaluating definite integrals.
Troubleshooting Tips/Error Traps:
- Students often make algebraic mistakes when finding antiderivatives. Students should get in the habit of differentiating their answer to verify that they found the correct antiderivative.
- In finding the area of a region between a curve and the x-axis, some students neglect to pay attention to the sign of the function. Functions that have both positive and negative values warrant special consideration. The region should be divided so that intervals for which the curve is above the x-axis and below the x-axis are considered separately. For any portion of the region below the x-axis, the value of the integral will be negative and is negated to represent the area. A good way to check results for this kind of problem is to find NINT (|f(x)|, x, a, b). This procedure is stated explicitly in Section 5.4, page 284.