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Unformatted text preview: (3/20/08) CHAPTER 6 INTEGRALS AND APPLICATIONS The derivative, which we studied in Chapters 2 through 5, is used to find rates of change of functions. In this chapter we begin the study of the second main tool of calculus, the integral , which is used to determine changes in values of quantities from their rates of change, to find areas, volumes, weights, average values, lengths of curves, and in many other applications. Section 6.1 gives a preview of the definition of the integral and of the Fundamental Theorem of Calculus in the context of functions whose rates of change are step functions. The definite integral is defined and some of its properties are discussed in Section 6.2/ In Section 6.3 we derive Part I of the Fundamental Theorem of Calculus , which deals with integrals of derivatives. Part II of the Fundamental Theorem concerning derivatives of definite integrals with variable endpoints is discussed in Section 6.4. The Fundamental Theorem is used in Section 6.5 to obtain a formula for definite integrals of power functions y = x n with constant n negationslash =- 1. In this section we also introduce the term indefinite integral for antiderivative. Section 6.6 deals with finding approximate values of definite integrals of functions given by graphs and tables. Section 6.7 covers integration formulas derived from differentiation formulas for transcendental functions. The technique of integration by substitution is discussed in Section 6.8. Section 6.1 Step function rates of change Overview: In this section we first look at applications where changes in values of functions can be determined from their rates of change without calculus because the rates of change are step functions. Then we obtain a preview of the definition of definite integrals (Section 6.2) and of Part I of the Fundamental Theorem of Calculus (Section 6.3) by applying the techniques of this section to approximations of continuous rates of change by step functions. Topics: Step function rates of change Approximating continuous rates of change with step functions Step function rates of change We begin with two examples that use the formula for distance traveled at constant velocity, [Distance traveled] = [Velocity] [Time] . Example 1 The step function v = v ( t ) of Figure 1 gives a mathematical model of the velocity of a bus. The bus travels 50 miles per hour for two hours, 25 miles per hour for one hour, and 75 miles per hour for three more hours. (a) How far does it travel in the entire six hours? (b) How is the answer to part (a) related to the area of the rectangles in Figure 2? Instructors who want to cover Part II of the Fundamental Theorem (derivatives of integrals) before Part I (integrals of derivatives) should cover Section 6.4 before Section 6.3....
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