Mathematics
Integrating Polynomials
Integrating polynomials involves finding the antiderivative of a polynomial function. This process requires applying the power rule for integration, which involves raising the exponent by 1 and dividing by the new exponent. The result is a new polynomial function with a constant of integration added.
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5 Key excerpts on "Integrating Polynomials"
eBook - PDF- David Pearson(Author)
- 1995(Publication Date)
- Butterworth-Heinemann(Publisher)
The process of finding all of the antiderivatives of a given function is the process, or operation, of integration. Integration, the determination of indefinite integrals, is the inverse operation to differentiation. Any general property of the derivative will also tell us something about the integral. It is therefore not particularly surprising that the three methods of integration which I shall present here, namely (i) integration by parts, (ii) integration by substitution or change of variable, and (iii) integration by the method of partial fractions, are in turn closely connected with results or ideas that I have already talked about with respect to differentiation; The Antiderivative 117 these are (i) the product rule, (ii) the chain rule, and (iii) the linearity of differentiati on. You may have noticed, perhaps in our treatment of the function e-x 2 in Section 8.1, a limitation in what can be achieved in the way of integration, which does not apply in the case of differentiation.You may, by now, feel confident enough to attempt the differentiation of just about any function which can be expressed in a simple way in terms of powers, exponentials, trigonometric functions and all the other functions which you use daily in mathematics theory and applications. You may not feel so confident with regard to integration. There are quite simple functions, of which e-x 2 is one, but cosix), sin xt x and ylln(x) are others which can be differentiated quite easily, but which have no integral which can be expressed in closed form in terms of the basic functions of calculus. On any interval on which these functions are defined and continuous, the indefinite integral does exist; it is just that the integral can only be expressed as a limit that cannot be evaluated in closed form, or as a series for which the sum is unknown, and so on. Nevertheless, I hope that you will not be deterred from approaching integration in a positive spirit.
eBook - PDFQuick Calculus
A Self-Teaching Guide
- Daniel Kleppner, Peter Dourmashkin, Norman Ramsey(Authors)
- 2022(Publication Date)
- Jossey-Bass(Publisher)
CHAPTER THREE Integral Calculus In this chapter you will learn about: • Antiderivatives and the indefinite integral; • Integrating a variety of functions; • Some applications of integral calculus; • Finding the area under a curve; • Definite integrals with applications; • Multiple integrals. The previous chapter was devoted to the first major branch of calculus—differential calculus. This chapter is devoted to the second major branch—integral calculus. The two branches have different natures: differential calculus has procedures that make it possible to differentiate any continuous function; integral calculus has no such general procedures—every problem presents a fresh puzzle. Nevertheless, integral calculus is essential to all of the sci- ences, engineering, economics, and in fact to every discipline that deals with quantitative information. There are two routes to introducing the concepts of integration. Although they start in different directions, they finally meld and create a single entity. If they were marked by road signs, the first would be “Antiderivatives and the indefinite integral” while the second would be “Area under a curve and the definite integral.” 169 170 Integral Calculus Chap. 3 3.1 Antiderivative, Integration, and the Indefinite Integral 306 The Antiderivative: The goal of this section is to learn some techniques for integration, sometimes called anti- differentiation. In this section we generally designate a function by f (x). The concept of the antiderivative is fundamental to the process of integration and is easily explained. When a function F (x) is differentiated to give f (x) = dF∕dx, then F (x) is an antiderivative of f (x), that is, F ′ (x) = f (x). This notation describes the defining property of the antiderivative, although only in terms of the derivative F ′ (x), not F (x) itself. The antiderivative is usually written in the form: F (x) = ∫ f (x)dx. The expression ∫ f (x) dx is also called the integral of f (x).
eBook - PDF- Sebastian J. Schreiber, Karl J. Smith, Wayne M. Getz(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
A systematic method for estimating the area under the curve was devised by Riemann, one of the great mathematicians of the nineteenth century. This method is commonly known as the Riemann sum and yields in the limit an object called the definite integral. The fathers of calculus, Newton and Leibniz, proved a connection between the problem of finding antiderivatives and finding areas under a curve. This connection, the fundamental theorem of calculus, which is presented in Section 5.4, helps make calculus one of the most powerful mathematical tools for understanding biological and physical processes. In Sections 5.5 through 5.7, we provide a short apprenticeship in various techniques used to compute and approximate integrals. Armed with these techniques, the chapter concludes with applications to cardiac output, survival and renewal equations, and the scientific notion of work. 347 348 Chapter 5 Integration 5.1 Antiderivatives Many mathematical operations have an inverse. For example, to undo the addition of b to a we subtract b: a + b − b = a. To undo division of a by b we multiply by b: a b b = a. To undo exponentiation, we take logarithms: ln e a = a. The process of differentiation can be undone by a process called antidifferentiation. To motivate antidifferentiation, consider how long it takes an organism to de- velop when the rate of development depends on environmental factors such as heat, light, and humidity. For example, the developmental rate of plants and insects, which lack internal thermal regulation mechanisms, depend critically on ambient tempera- ture. For ambient temperatures within a range defined by developmental thresholds, a plant’s or insect’s organismal developmental rate can often be approximated by an increasing linear function of temperature.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
The founders of the calculus thought of the integral as an infinite sum of rectangles of infinitesimal width. A rigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a limiting procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs. Beginning in the nineteenth century, more sophisticated notions of integrals began to appear, where the type of the function as well as the domain over which the integration is performed has been generalised. A line integral is defined for functions of two or three variables, and the interval of integration [ a , b ] is replaced by a certain curve connecting two points on the plane or in the space. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space. Integrals of differential forms play a fundamental role in modern differential geometry. These generalizations of integrals first arose from the needs of physics, and they play an important role in the formulation of many physical laws, notably those of electrodynamics. There are many modern concepts of integration, among these, the most common is based on the abstract mathematical theory known as Lebesgue integration, developed by Henri Lebesgue. History Pre-calculus integration Integration can be traced as far back as ancient Egypt ca. 1800 BC, with the Moscow Mathematical Papyrus demonstrating knowledge of a formula for the volume of a pyramidal frustum. The first documented systematic technique capable of determining integrals is the method of exhaustion of the ancient Greek astronomer Eudoxus ( ca. 370 ________________________ WORLD TECHNOLOGIES ________________________ BC), which sought to find areas and volumes by breaking them up into an infinite number of shapes for which the area or volume was known.
eBook - PDF- Daniel Ashlock(Author)
- 2022(Publication Date)
- Springer(Publisher)
1 C H A P T E R 1 Integration, Area, and Initial Value Problems The two main concepts in calculus are derivatives and integrals. We assume you have already studied derivatives. This chapter introduces the concept of integrals. Derivatives and integrals were developed independently, and later it was discovered that they are closely related. We will start with this relationship and define integrals as a sort of backward derivative. Then, we will in- troduce some of the primary uses of integration – finding areas under curves and solving bound- ary value problems. Finally, we will show how integrals were originally defined. This chapter does not cover how to integrate most functions. That we’ve saved for later in Chapter 4. 1.1 ANTI-DERIVATIVES The anti-derivative of a function is another function whose derivative is the function you started with. More precisely: Definition 1.1 If f .x/ and F.x/ are functions so that F 0 .x/ D f .x/; we say that F.x/ is an anti-derivative of f .x/. The terminology anti-derivative is fairly modern. The original terminology is to call an anti- derivative of f .x/ an integral of f .x/. Integrals also have a special notation. Knowledge Box 1.1 Integral notation If F.x/ is an anti-derivative of f .x/, we write F.x/ D Z f .x/ dx and call F.x/ an integral of f .x/. 2 1. INTEGRATION, AREA, AND INITIAL VALUE PROBLEMS Example 1.1 Find an anti-derivative of f .x/ D 2x. Solution: We know that the derivative of F.x/ D x 2 is f .x/ D 2x. So, one possible answer is: F.x/ D x 2 ˙ One of the issues that arises when we work with anti-derivatives is that they are not unique. The derivative of G.x/ D x 2 C 5 is also 2x. This means that anti-derivatives (integrals) are known only up to some constant value. We will develop techniques for dealing with this in Section 1.3, but for now we will simply use unknown constants or constants of integration. Example 1.2 Use integral notation, with an unknown constant C , to represent all the anti-derivatives of 2x.
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