Approximating Areas

4 Key excerpts on "Approximating Areas"

• 

  eBook - PDF

  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).

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Journey into Mathematics

  An Introduction to Proofs

  • Joseph J. Rotman(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  Chapter 3

  Circles and π

  APPROXIMATIONS

  “All right,” said the Cat; and this time it vanished quite slowly, beginning with the end of the tail, and ending with the grin, which remained some time after the rest of it had gone.

  Lewis Carroll, Alice in Wonderland

  Having seen how to compute the area of triangles, parallelograms, and polygons, we are now going to consider curved regions. Areas of polygons can be found by first subdividing them into squares or triangles and then adding up the areas of each of the individual pieces. But not every region can be so subdivided; for example, a disk (i.e., a bounded region in the plane whose circumference is a circle) cannot be subdivided into (a finite number of) polygons. The basic idea, now, is to find the exact area of a disk by approximating it by the areas of inscribed polygons. We will give more details below, but the reader should now realize why the discussion of the area of a disk will have a different flavor than that of our earlier discussion of areas.

  Let us begin by seeing how one might have discovered the formula for the area of a disk. Let D be a disk with radius r and circumference c. Our goal is to find the usual area formula

  (if c = 2πr , then ).

  Figure 3.1

  Divide D into 4 equal sectors and rearrange them in a row:

  Figure 3.2

  Of course, the area remains unchanged, and the total length of the top 4 arcs is still c . Now double the area by adding 4 shaded sectors.

  Figure 3.3

  If we denote area(D ) by A , then the area of this new figure is 2A . It looks a bit like a parallelogram: each of the scalloped top and bottom edges has length c, and each of the two side edges has length r (for they are radii of D ). Now divide D

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Anton's Calculus

  Early Transcendentals

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  253 5 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 functions that can be defined using integration, with a focus on the natural logarithm function. INTEGRATION 5.1 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 mathematicians. The first real progress in dealing with the general area problem was made by the Greek mathematician 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 exhaustion. The method, when applied to a circle, consists of inscrib- ing a succession of regular polygons in the circle and allowing the number of sides to in- crease indefinitely (Figure 5.1.1). As the number of sides increases, the polygons tend to “exhaust” the region inside the circle, and the areas of the polygons become better and better approximations of the exact area of the circle.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Calculus

  Early Transcendental Single Variable

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  The method, when applied to a circle, consists of in- scribing a succession of regular polygons in the circle and allowing the number of sides to increase indefinitely (Figure 5.1.1). As the number of sides increases, the polygons tend to “exhaust” the region inside the circle, and the areas of the polygons become better and better approximations of the exact area of the circle. To see how this works numerically, let A(n) denote the area of a regular n-sided polygon inscribed in a circle of radius 1. Table 5.1.1 shows the values of A(n) for various choices of n. Note that for large values of n the area A(n) appears to be close to π (square units), as one would expect. This suggests that for a circle of radius 1, the method of exhaustion is equivalent to an equation of the form lim n →∞ A(n) = π Since Greek mathematicians were suspicious of the concept of “infinity,” they avoided its use in mathematical arguments. As a result, computation of area using the method of exhaustion was a very cumbersome procedure. It remained for Newton and Leibniz to obtain a general method for finding areas that explicitly used the notion of a limit. We will discuss their method in the context of the following problem. 254 Chapter 5 / Integration Figure 5.1.1 Table 5.1.1 5.1.1 THE AREA PROBLEM Given a function f that is continuous and nonnegative on an interval [a, b], find the area A of the region R that lies between the graph of f and the interval [a, b] on the x-axis (Figure 5.1.2). Figure 5.1.2 THE RECTANGLE METHOD FOR FINDING AREAS One approach to the area problem is to use Archimedes’ method of exhaustion in the following way: Figure 5.1.3 Logically speaking, we cannot really talk about computing areas without a precise mathematical definition of the term “area.” Later in this chapter we will give such a definition, but for now we will treat the concept intuitively.

  Sign up to read

  Learn more about book