Integration

3 Key excerpts on "Integration"

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  Calculus

  Single Variable

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  200 CHAPTER 4 kata716/iStock/Getty Images The change in size of a minnow population can be obtained by integrating the population’s growth rate over the relevant time interval. Integration In this chapter we will introduce “Integration,” a process motivated by the problem of computing the area of plane regions. After an informal overview of the problem, we will discuss a surprising relationship between Integration and differentiation that is known as the Fundamental Theorem of Calculus. We will then apply Integration to continue our study of rectilinear motion and to define the “average value" of a function. We conclude the chapter by studying some consequences of the chain rule in integral calculus. 4.1 An Overview of the Area Problem In this introductory section we will consider the problem of calculating areas of plane regions with curvilinear boundaries. All of the results in this section will be reexamined in more detail later in this chapter. Our purpose here is simply to introduce and motivate the fundamental concepts. The Area Problem Formulas for the areas of polygons, such as squares, rectangles, triangles, and trapezoids, were well known in many early civilizations. However, the problem of finding formulas for regions with curved boundaries (a circle being the simplest example) caused difficulties for early mathe- maticians. The first real progress in dealing with the general area problem was made by the Greek math- ematician Archimedes, who obtained areas of regions bounded by circular arcs, parabolas, spirals, and various other curves using an ingenious procedure that was later called the method of exhaus- tion. The method, when applied to a circle, consists of inscribing a succession of regular polygons in the circle and allowing the number of sides to increase indefinitely (Figure 4.1.1).

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  Calculus Early Transcendentals

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  Differentiation and Integration are Inverse Processes The two parts of the Fundamental Theorem of Calculus, when taken together, tell us that differentiation and Integration are inverse processes in the sense that each undoes the effect of the other. To see why this is so, note that Part 1 of the Fundamental Theorem of Calculus (5.6.1) implies that ∫ x a f ′ (t) dt = f(x) − f(a) which tells us that if the value of f (a) is known, then the function f can be recovered from its derivative f ′ by integrating. Conversely, Part 2 of the Fundamental Theorem of Calculus (5.6.3) states that d dx [ ∫ x a f(t) dt ] = f(x) which tells us that the function f can be recovered from its integral by differentiating. Thus, differentiation and Integration can be viewed as inverse processes. It is common to treat parts 1 and 2 of the Fundamental Theorem of Calculus as a single theorem and refer to it simply as the Fundamental Theorem of Calculus. This theorem ranks as one of the greatest discoveries in the history of science, and its formulation by Newton and Leibniz is generally regarded to be the “discovery of calculus.” Integrating Rates of Change The Fundamental Theorem of Calculus ∫ b a f(x) dx = F(b) − F(a) (14) has a useful interpretation that can be seen by rewriting it in a slightly different form. Since F is an antiderivative of f on the interval [a, b], we can use the relationship F ′ (x) = f(x) to rewrite (14) as ∫ b a F ′ (x) dx = F(b) − F(a) (15) 332 Chapter 5 / Integration In this formula we can view F ′ (x) as the rate of change of F(x) with respect to x, and we can view F(b) − F(a) as the change in the value of F(x) as x increases from a to b (Figure 5.6.11). Thus, we have the following useful principle. a b y = F(x) Slope = F ′(x) x y F(b) − F(a) Integrating the slope of y = F(x) over the interval [a, b] produces the change F(b) − F(a) in the value of F(x).

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  Introduction to Integral Calculus

  Systematic Studies with Engineering Applications for Beginners

  • Ulrich L. Rohde, G. C. Jain, Ajay K. Poddar, A. K. Ghosh(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  Later on, the concept of the definite integral was also developed. Newton and Leibniz recognized the importance of the fact that finding derivatives and finding integrals (i.e., antiderivatives) are inverse processes, thus making possible the rule for evaluating definite integrals. All these matters are systematically introduced in Part II of the book. (There were many difficulties in the foundation of the subject of Calculus. Some problems reflecting conflicts and doubts on the soundness of the subject are reflected in the “Historical Notes” given at the end of Chapter 9 of Part I.) During the last 150 years, Calculus has matured bit by bit. In the middle of the nineteenth century, French Mathematician Augustin-Louis Cauchy (1789–1857) gave the definition of limit, which removed all doubts about the soundness of Calculus and xvi PREFACE made it free from all confusion. It was then that Calculus had become, mathematically, much as we know it today. To obtain the derivative of a given function (and to apply it for studying the properties of the function) is the subject of the ‘differential calculus’. On the other hand, computing a function whose derivative is the given function is the subject of integral calculus. [The function so obtained is called an anti-derivative of the given function.] In the operation of computing the antiderivative, the concept of limit is involved indirectly. On the other hand, in defining the definite integral of a function, the concept of limit enters the process directly. Thus, the concept of limit is involved in both, differential and integral calculus. In fact, we might define calculus as the study of limits. It is therefore important that we have a deep understanding of this concept. Although, the topic of limit is rather theoretical in nature, it has been presented and discussed in a very simple way, in the Chapters 7(a) and 7(b) of Part-I (i.e. Differential Calculus) and in Chapter 5 of Part-II (i.e. Integral Calculus).

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