Mathematics
Improper Integrals
Improper integrals are integrals that do not converge to a finite value. They arise when the function being integrated is unbounded or the interval of integration is infinite. Improper integrals can be evaluated using limits or by breaking up the integral into smaller pieces.
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10 Key excerpts on "Improper Integrals"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 3 Improper Integral and Multiple Integral Improper integral An improper integral of the first kind. The integral may need to be defined on an unbou-nded domain. ________________________ WORLD TECHNOLOGIES ________________________ An improper Riemann integral of the second kind. The integral may fail to exist because of a vertical asymptote in the function. In calculus, an improper integral is the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞ or, in some cases, as both endpoints approach limits. Specifically, an improper integral is a limit of the form or of the form in which one takes a limit in one or the other (or sometimes both) endpoints (Apostol 1967, §10.23). Integrals are also improper if the integrand is undefined at an interior point of the domain of integration, or at multiple such points. It is often necessary to use Improper Integrals in order to compute a value for integrals which may not exist in the conventional sense (as a Riemann integral, for instance) ________________________ WORLD TECHNOLOGIES ________________________ because of a singularity in the function, or an infinite endpoint of the domain of inte-gration. Examples The following integral does not exist as a Riemann integral because the domain of integration is unbounded. (The Riemann integral is only well-defined over a bounded domain.) However, it may be assigned a value as an improper integral by interpreting it instead as a limit The following integral also fails to exist as a Riemann integral: Here the function is unbounded, and the Riemann integral is not well-defined for unbo-unded functions. However, if the integral is instead understood as the limit: then the limit converges. Convergence of the integral An improper integral converges if the limit defining it exists.
- Prem K. Kythe, Michael R. Schäferkotter(Authors)
- 2004(Publication Date)
- Chapman and Hall/CRC(Publisher)
4 Improper Integrals Improper Integrals are such that their range or integrands are unbounded. They are defined as the limits of certain proper integrals. Thus, an improper integral over the interval [0 , ∞ ) is defined as ∞ 0 f ( x ) dx = lim R →∞ R 0 f ( x ) dx whenever the limit exists. Improper Integrals over [ a, ∞ ) , ( −∞ , a ] are similarly defined. Improper Integrals over the interval ( −∞ , ∞ ) are defined in two ways: First, there is the usual definition: ∞ −∞ f ( x ) dx = 0 −∞ f ( x ) dx + ∞ 0 f ( x ) dx . Then, there is the other definition: ∞ −∞ f ( x ) dx = lim R →∞ R − R f ( x ) dx , provided both limits exist. This definition is also known as Cauchy’s principal value (or p.v.) of the integral, denoted by − ∞ −∞ f ( x ) dx . A common p.v. integral is the Hilbert transform − b A f ( t ) t − x dt , where −∞ ≤ a < b ≤ ∞ , and a < x < b . A sufficient condition for the Hilbert transform to exist over a finite interval [ a, b ] is that f ( t ) satisfy a Lipschitz or H¨older condition in [ a, b ] ; i.e., there are constants A > 0 and 0 < α ≤ 1 such that for any two points t 1 , t 2 ∈ [ a, b ] we have f ( t 1 ) − f ( t 2 ) ≤ A t 1 − t 2 α . Cauchy’s p.v. integrals are discussed in Chapter 6. 4.1. Infinite Range Integrals We discuss some methods and quadrature rules for computation of infinite range integrals. 4.1.1. Truncation Method. The technique of truncating the infinite interval of integration is discussed, and the Riemann method is used on the truncated interval, 187 188 4. Improper Integrals which is valid if the integrand decays rapidly. We consider infinite range integrals of the form I 1 = ∞ a w ( x ) f ( x ) dx, I 2 = ∞ −∞ w ( x ) f ( x ) dx.
eBook - PDF- Ron Larson, Bruce Edwards(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
CONCEPT CHECK 1. Improper Integrals Describe two ways for an integral to be improper. 2. Improper Integrals What does it mean for an improper integral to converge? 3. Indefinite Integration Limits Explain how to evaluate an improper integral that has an infinite limit of integration. 4. Finding Values For what values of a is each integral improper? Explain. (a) integral.alt1 5 a 1 x + 2 dx (b) integral.alt1 4 a x 3x - 1 dx Determining Whether an Integral Is Improper In Exercises 5–12, decide whether the integral is improper. Explain your reasoning. 5. integral.alt1 1 0 dx 5x - 3 6. integral.alt1 2 1 dx x 3 7. integral.alt1 1 0 2x - 5 x 2 - 5x + 6 dx 8. integral.alt1 ∞ 1 ln x 2 dx 9. integral.alt1 2 0 e -x dx 10. integral.alt1 ∞ 0 cos x dx 11. integral.alt1 ∞ -∞ sin x 4 + x 2 dx 12. integral.alt1 πH208624 0 csc x dx Evaluating an Improper Integral In Exercises 13–16, explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converges. 13. integral.alt1 4 0 1 radical.alt2x dx 14. integral.alt1 4 3 1 (x - 3) 3H208622 dx x 1 1 2 2 4 4 3 3 y x 1 2 4 5 10 20 40 30 50 y 15. integral.alt1 2 0 1 (x - 1) 2 dx 16. integral.alt1 0 -∞ e 3x dx x 1 2 2 y y x - 1 1 Evaluating an Improper Integral In Exercises 17–32, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. 17. integral.alt1 ∞ 2 1 x 3 dx 18. integral.alt1 ∞ 3 1 (x - 1) 4 dx 19. integral.alt1 ∞ 1 3 3 radical.alt2x dx 20. integral.alt1 ∞ 1 4 4 radical.alt2x dx 21. integral.alt1 ∞ 0 e xH208623 dx 22. integral.alt1 0 -∞ xe -4x dx 23. integral.alt1 ∞ 0 x 2 e -x dx 24. integral.alt1 ∞ 0 e -x cos x dx 25. integral.alt1 ∞ 4 1 x(ln x) 3 dx 26. integral.alt1 ∞ 1 ln x x dx 27. integral.alt1 ∞ -∞ 4 16 + x 2 dx 28. integral.alt1 ∞ 0 x 3 (x 2 + 1) 2 dx 29. integral.alt1 ∞ 0 1 e x + e -x dx 30. integral.alt1 ∞ 0 e x 1 + e x dx 31. integral.alt1 ∞ 0 cos πx dx 32.
eBook - PDFCalculus
Single Variable
- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2022(Publication Date)
- Wiley(Publisher)
We will call the vertical asymptotes infinite discontinuities, and we will call integrals with infinite intervals of integration or infinite discontinuities within the interval of integration Improper Integrals. Here are some examples: • Improper Integrals with infinite intervals of integration: +∞ 1 dx x 2 , 0 −∞ e x dx, +∞ −∞ dx 1 + x 2 • Improper Integrals with infinite discontinuities in the interval of integration: 3 −3 dx x 2 , 2 1 dx x − 1 , π 0 tan x dx • Improper Integrals with infinite discontinuities and infinite intervals of integration: +∞ 0 dx √ x , +∞ −∞ dx x 2 − 9 , +∞ 1 sec x dx Integrals Over Infinite Intervals To motivate a reasonable definition for Improper Integrals of the form +∞ a f (x) dx let us begin with the case where f is continuous and nonnegative on [a, +∞), so we can think of the integral as the area under the curve y = f (x) over the interval [a, +∞) (Figure 7.8.1). At x y a +∞ a f (x) dx FIGURE 7.8.1 first, you might be inclined to argue that this area is infinite because the region has infinite extent. However, such an argument would be based on vague intuition rather than precise mathematical logic, since the concept of area has only been defined over intervals of finite extent. Thus, before we can make any reasonable statements about the area of the region in Figure 7.8.1, we need to begin by defining what we mean by the area of this region. For that purpose, it will help to focus on a specific example. Suppose we are interested in the area A of the region that lies below the curve y = 1 / x 2 and above the interval [1, +∞) on the x-axis. Instead of trying to find the entire area at once, let us begin by calculating the portion of the area that lies above a finite interval [1, b], where b > 1 is arbitrary. That area is b 1 dx x 2 = − 1 x b 1 = 1 − 1 b (Figure 7.8.2).
No longer available |Learn more- Ioannis Roussos(Author)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 1 Improper Riemann Integrals 1.1 Definitions and Examples Many theorems in Mathematics and many applications in science and technology depend on the evaluation and on the properties of improper Riemann integrals. Therefore, we are going to state the definitions of im-proper or generalized integrals and then discuss their properties. Subse-quently, we discuss criteria for checking their existence (or non-existence) and then we develop methods and mathematical techniques we can use in order to evaluate them. Certainly the answers to many important improper Riemann integrals have been tabulated in mathematical hand-books and can also be found with the help of various computer programs, which we can use if we can trust in them, of course. However, these means can never exhaust every interesting case. Hence, the good knowledge of the mathematical theory of how to understand, handle and compute im-proper integrals, at a higher level, will always remain very important for being able to deal with new cases and checking the accuracy of the answers provided in tables or found by computer programs or packages. In a regular undergraduate Calculus course we study the Fundamen-tal Theorem of Integral Calculus. This states: Theorem 1.1.1 (Fundamental Theorem of Integral Calculus) If a real function f : [ a, b ] → R ( a < b are real numbers) is continuous, then it possesses antiderivatives F ( x ) , i.e., functions that satisfy F ( x ) = f ( x ) for every x ∈ [ a, b ] [at the end points we consider the appropriate side derivatives, F + ( a ) = f ( a ) and F − ( b ) = f ( b ) ]. Any such antiderivative F ( x ) of f ( x ) is necessarily continuous in ( a, b ) , right continuous at a , [ i.e., F ( a ) = lim x → a + F ( x ) ] , left continuous at b , [ i.e., F ( b ) = lim x → b − F ( x ) ] , and satisfies b a f ( x ) dx = F ( b ) − F ( a ) . 1 2 Improper Riemann Integrals We emphasize the three hypotheses that must hold in order for this Theorem to be valid: 1.
eBook - PDFThe Calculus Lifesaver
All the Tools You Need to Excel at Calculus
- Adrian Banner(Author)
- 2009(Publication Date)
- Princeton University Press(Publisher)
C h a p te r 20 Improper Integrals: Basic Concepts This is a difficult topic, so I’m devoting two chapters to it. This chapter serves as an introduction to Improper Integrals. The next chapter gets into the details of how to solve problems involving Improper Integrals. If you are reading this chapter for the first time, you should probably take care to try to understand all the points in it. On the other hand, if you are reviewing for a test, most likely you’ll want to skim over the chapter, noting the boxed formulas and the sections marked as important, and concentrate on the next chapter. Here’s what we’ll actually look at in this chapter: • the definition of Improper Integrals, convergence, and divergence; • Improper Integrals over unbounded regions; and • the theoretical basis for the comparison test, the limit comparison test, the p -test, and the absolute convergence test. We’ll revisit all four of these tests in the next chapter and see many examples of how to apply them. 20.1 Convergence and Divergence What is an improper integral, anyway? In Chapter 16, we saw that the integral Z b a f ( x ) dx certainly makes sense if f is a bounded function on [ a, b ] which is continuous except at a finite number of places. If f has infinitely many discontinuities, the integral might still make sense, or it might be totally screwed up (see Section 16.7 of Chapter 16 for an example). What if f isn’t bounded? This means that the values of f ( x ) manage to get really large (positively or neg-atively or both) while x is in the interval [ a, b ]. This sort of thing typically happens when f has a vertical asymptote somewhere in this interval: the function blows up there and can’t be bounded. This causes the above integral to be improper. 432 • Improper Integrals: Basic Concepts There’s a different type of unboundedness that can occur even if f is bounded. The interval [ a, b ] can actually be infinite—something like [0 , ∞ ), [ -7 , ∞ ), ( -∞ , 3] or even ( -∞ , ∞ ).
eBook - PDF- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2022(Publication Date)
- Wiley(Publisher)
Thus, for example, a function with a vertical asymptote within the interval of integration would not be integrable. Our main objective in this section is to extend the concept of a definite integral to allow for infinite intervals of integration and integrands with vertical asymptotes within the interval of integration. We will call the vertical asymptotes infinite discontinuities, and we will call integrals with infinite intervals of integration or infinite discontinuities within the interval of integration Improper Integrals. Here are some examples: • Improper Integrals with infinite intervals of integration: ∫ +∞ 1 dx x 2 , ∫ 0 −∞ e x dx, ∫ +∞ −∞ dx 1 + x 2 • Improper Integrals with infinite discontinuities in the interval of integration: ∫ 3 −3 dx x 2 , ∫ 2 1 dx x − 1 , ∫ 0 tan x dx • Improper Integrals with infinite discontinuities and infinite intervals of integration: ∫ +∞ 0 dx √ x , ∫ +∞ −∞ dx x 2 − 9 , ∫ +∞ 1 sec x dx Integrals Over Infinite Intervals To motivate a reasonable definition for Improper Integrals of the form ∫ +∞ a f(x) dx x y a +∞ a f (x) dx ▴ Figure 7.8.1 let us begin with the case where f is continuous and nonnegative on [a, +∞), so we can think of the integral as the area under the curve y = f(x) over the interval [a, +∞) (Figure 7.8.1). At first, you might be inclined to argue that this area is infinite because the region has infinite 510 Chapter 7 / Principles of Integral Evaluation extent. However, such an argument would be based on vague intuition rather than precise mathematical logic, since the concept of area has only been defined over intervals of finite extent. Thus, before we can make any reasonable statements about the area of the region in Figure 7.8.1, we need to begin by defining what we mean by the area of this region. For that purpose, it will help to focus on a specific example.
eBook - PDF- G. M. Fikhtengol'ts, I. N. Sneddon, M. Stark, S. Ulam(Authors)
- 2014(Publication Date)
- Pergamon(Publisher)
CHAPTER 17 I M P R O P E R I N T E G R A L S § 1. Improper Integrals with infinite limits 282. The definition of integrals with infinite limits. In Chapter 11 b we studied the concept of definite integrals f{x)dx for the case α of a finite interval [a, b] and a bounded function f(x). The present chapter is devoted to a generalization of this concept in several directions. We begin by considering an integral extended over an infinite interval. Let the function f(x) be defined over the interval [a, oo), i.e. for Λ : > a , and integrable over any finite portion of it [a. A], so that A the integral f(pc)dx has a meaning for any A>a, a The finite or infinite limit of this integral as A-^ oo is called the (improper) integral of the function f(x) over the interval from a to 0 0 and it is denoted by the symbol 00 A f(x)dx=Umf(x)dx, (1) a a In the case when there exists a finite limit, we say that the integral (1) converges, and the function f(pc) is said to be integrable in the infinite interval [a, oo]. The only difference from the integral in the proper sense, or proper integral, studied previously is that the integral (1) defined is called improper t. If the limit (1) is infinite or does not exist at all, then one says of the integral that it diverges. t We recall that we have already encountered the concept of an improper integral in Sec. 241. [110] § 1. INTEGRALS WITH INFINITE LIMITS 111 J l+x'^ ^ -. 0 0 J 2 t We exclude only the case when these integrals are equal to an infinity of different values. Analogously to ( 1 ) , we define the integral of the function f(x) from — CO to a: a a lf(x)dx= lim lf(x)dx(A'
- Luis Manuel Braga da Costa Campos(Author)
- 2010(Publication Date)
- CRC Press(Publisher)
17 Improper Integrals and Principal Value A real integral is a particular case of a contour integral, in that the path of integration lies along the real axis, that is, it consists of one or several straight segments. Complex integrals along open contours in general, and real integrals in particular, can be calculated by means of residues, by two methods: (i) using a change of variable to transform the open contour into a closed loop (Section 17.2); (ii) extension of the open loop until it becomes a closed loop, in such a way (Section 17.3) that the contribution to the integral of the additional path can be calculated (or made to vanish). In both cases (i) and (ii), a loop integral results that may be calculated by residues (Chapter 15). The Riemann integral defined before (Chapter 13) is a proper integral, as it applies to a bounded function over a finite path (Section 17.1). The integral is improper of the first (second) kind if the path is infinite (function is unbounded), and of mixed kind is both features occur. A improper integral of first kind (Sections 17.4 and 17.5) is unilateral (bilateral) if only one (both) of the end-points are at infinity; an improper integral of the second kind (Section 17.6) is unilateral (bilateral) if the singularity of the integrand occurs at the end-points (in the interior of the path of integration). Improper Integrals are calculated as limits of proper integrals, for example, as the end-point(s) tend to infinity (the integral approaches its singularity) for the first (second) kind. If the integrand is a many-valued function, with branch-point on the path of integration, the integral may become indeterminate, unless a principal value is chosen (Section 17.7). Even for a single-valued integrand, the primitive may be multi-valued, leading to an improper integral of the second kind whose value is indeterminate (Section 17.8), and can be specified uniquely as the Cauchy principal value (Section 17.9).
eBook - PDFBrief Calculus
An Applied Approach
- Michael Sullivan(Author)
- 2021(Publication Date)
- Wiley(Publisher)
The limits of integration a and bare both finite. 2. The nction is continuous on [a, b]. In many situations, one or both of these assumptions are not met. For example, one of the limits of integration might be infinity; or the nction y = f(x) might be discontin- uous at some number in [a, b]. If either of the conditions ( 1) and (2) are not satisfied, then f f(x) dx is called an improper integral. One Limit of Integration Is Infinite We begin with an example. Finding an Area Find the area under the graph of f(x) = � to the right of x = 1. X SOLUTION First we graph f(x) = �- See Figure 1. The area to the right of x = 1 is shaded. To find X this area, we pick a number b to the right of x = 1. The area under the graph of f(x) = � om x = 1 to x = b is X J b �dx =(-)lb= _ + X X I b 1 (1, 1 � This area depends on the choice of b. Now the area we seek is obtained by letting b - oo. Since 3 b EXAMPLE 2 Improper Integrals 487 we conclude that the area under the graph off(x) = to the right of x = 1 is 1. t The area we und in Example 1 can be represented symbolically by the improper integral and it can be evaluated by finding J oo 1 -dx x2 f b 1 Jim - 2 dx boo X This leads us to rmulate the llowing definition. Improper Integral Suppose a nction f is continuous on the interval [ a, oo). The improper tegral, L f(x) , is defined as { f(x) dx = Jim { b f(x) J n boo J a provided this limit exists and is a real number. If this limit does not exist or if it is infinite, the improper integral has no value. Suppose a nctionfis continuous on the interval (-oo, b]. The improper integral, f J (x) dx, is defined as f j (x) dx = E� f f(x) provided that this limit exists and is a real number. If this limit does not exist or if it is infinite, the improper integral has no value. Evaluating Improper Integrals Find the value, if there is one, of J o f 1 (a) e x (b) - dx -oo I X SOLUTION (a) f e x d x = a � � f e x dx = .
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