Documents about Improper Integrals
Lecture_4
RPI, MATH 1020
Excerpt: ... CALCULUS II Spring 2009 LECTURE 4 This lecture provides eight examples of improper integrals . 1 Recall We recall from Lecture 3 every polynomial can (theoretically) be written as a product of linear and quadratic factors integration of rational functions can be eected by partial fraction decomposition of the integrand End of Recall 2 Mathematics Interlude I Improper Integrals We recall briey the denition of the Riemann Integral. Denition. Let f denote a real-valued function dened on the closed and bounded interval [a, b]. Let L denote a real number such that for each > 0 there exists > 0 such that if {xi }n is a partition of [a, b] with i=0 max1in xi < , then n f (i )xi L < i=1 for every choice of the i s constrained by xi1 i xi , 1 i n. The number L is called the integral of f , or the Riemann integral of f . One writes the integral of f by b f (x) dx. a If a real-valued function dened on [a, b] has a Riemann integral the ...
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28MoreImproperInt
Virginia Tech, MATH 1206
Excerpt: ... More Improper Integrals ...
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28MoreImproperInt
Virginia Tech, MATH 1206
Excerpt: ... More Improper Integrals ...
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Assgn6M212
Piedmont, MATH 212
Excerpt: ... Math 212 Spring 2009 Montgomery Homework 6 The following are due February 27, 2008. Pay special attention to writing up your solutions (especially proofs) before you submit the assignment. Start it early or face dire consequences! 1. Determine i ...
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intimp
Kennesaw, MATH 2202
Excerpt: ... Improper Integrals Dr. Philippe B. laval Kennesaw State University September 19, 2005 Abstract Notes on improper integrals . 1 1.1 Improper Integrals Introduction b In Calculus II, students dened the integral a f (x) dx over a nite interval [a, b]. The function f was assumed to be continuous, or at least bounded, otherwise the integral was not guaranteed to exist. Assuming an antiderivative of f could b be found, a f (x) dx always existed, and was a number. In this section, we investigate what happens when these conditions are not met. Denition 1 (Improper Integral) An integral is an improper integral if either the interval of integration is not nite (improper integral of type 1) or if the function to integrate is not continuous (not bounded) in the interval of integration (improper integral of type 2). Example 2 0 1 0 0 ex dx is an improper integral of type 1 since the upper limit of integration is innite. Example 3 ous at 0. Example 4 dx 1 is an improper integral of type 2 beca ...
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writing assignment two
Mesa CC, MATH 151
Excerpt: ... Scott B. Roberts Writing Assignment Two The phrase "improper integral" means that a definite integral contains intervals that are either infinite, or the function has infinite discontinuity. They are also known as an infinite integral. In other words, an integral is stated to be improper if A) The interval of integration is unbounded and, or B) The function f(x) has an infinite discontinuity at some point C in [a,b]. If the integral has an infinite interval, infinity or negative infinity, then the integral is said to be improper. Integrating a function that has a vertical asymptote, would mean that the integral contains infinite discontinuity. For example, if you have (x+2) in the denominator, than the integral is discontinuous at -2, which would cause the denominator to be zero. So this integral would be considered improper. We evaluate improper integrals by substituting in a dummy variable in place of the infinite interval, and then take the limit of the dummy variable as it approaches the infinite in ...
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m122_f08_rhandout4b
Carnegie Mellon, MATH 122
Excerpt: ... Math 122 Fall 2008 Handout 4(b): Improper Integrals through Algebraic Comparisons For each of the integrals given on this handout, determine whether the integral converges or diverges. You should not use your calculator to evaluate antiderivatives here. If you cant work out an antiderivative, try comparing the integral in question to an improper integral that is easier to work out (e.g. a p-integral or the integral of an exponential function). If you do use comparison to work out the convergence or divergence of any of the improper integrals , note down: (I) (II) (III) (IV) " Your initial guess concerning convergence or divergence. The improper integral that you plan to use for comparison. The work that shows that the integral from Step (II) is greater than or less than the integral you are investigating. Your final conclusion. (a) # 1+ x 3 1 3 2 dx . Converges Diverges Initial Guess: ! Improper Integral for Comparison: " (b) # 4 3 + sin( x ) dx . x Converges Diverges Initial Guess: ! Impr ...
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lecture8_8s
UC Davis, MATH 8
Excerpt: ... 8.8 Improper integrals Recall that in the study of denite integral b f (x) dx, a we assume that the interval [a, b] is nite the function f is bounded. Such integrals are said to be proper. How about the area under the following curves? This leads to improper integrals ! Improper integrals Type I: the interval is innite: [a, ), (, b] or (, ). Type II: the function f is unbounded. Idea: improper integrals are calculated as limits. Type I: Integral over innite intervals. First Prev Next Last Go Back Full Screen Close Quit 1. If f (x) is continuous on [a, ), then b f (x) dx = lim a b f (x) dx. a 2. If f (x) is continuous on (, b], then b b f (x) dx = lim a f (x) dx. a 3. If f (x) is continuous on (, ), then a f (x) dx = f (x) dx + a f (x) dx where a is any real number. In each case, if the limit is nite, we say the improper integral converges and that the limi ...
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311t2r
Rutgers, MATH 311
Excerpt: ... Math 311-Advanced CalculusTest 2 Review November, 2003 The best way to review for the test is to read the textbook and lecture notes and to do as many homework problems as possible. (1) Know the - definition of limxa f (x) = L. Know how to use the definition to prove a limit. See for example, problems 8, 10, 11 in Section 3.1. (2) Understand the proofs of theorems 3.2.1, 3.2.2, 3.2.5. Understand the difference between continuity and uniform continuity. See for example, problems 4, 9, 10 in Section 3.2. (3) Understand the concepts of upper partial sum, lower partial sum. Understand the concept of Riemann integrable, Riemann partial sum, and Riemann integral. Know how to estimate a Riemann integral by using Theorem 3.3.4. Understand the proof of the fact that a bounded continuous function with finitely many discontinuous points is Riemann integrable. See for example, problems 1, 3, 4, 10 in Section 3.3 and problems 1, 3 in Section 3.5. (4) Understand the definitions of improper integrals . Know how to determi ...
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c7s7bw
ETSU, MATH 1920
Excerpt: ... 7.7 Improper Integrals 1 Chapter 7. Integration Techniques, lHpitals o Rule, and Improper Integrals 7.7 Improper Integrals Note. In this section, we are interested in nding the area under a curve over an innite interval. This arises, in particular, in probability and statistics when looking at, for example, the area under the normal distribution. Denition. Integrals with innite limits of integration are improper integrals : 1. If f (x) is continuous on [a, ), then b f (x) dx = lim a b f (x) dx. a 2. If f (x) is continuous on (, b], then b b f (x) dx = lim a f (x) dx. a 3. If f (x) is continuous on (, ), then c f (x) dx = f (x) dx + c f (x) dx. 7.7 Improper Integrals 2 In parts 1 and 2, if the limit is nite, the improper integral converges and the limit is the value of the improper integral. If the limit fails to exist, the improper integral diverges. In part 3, the integral on the left-hand side of the e ...
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c7s7
ETSU, MATH 1920
Excerpt: ... 7.7 Improper Integrals 1 Chapter 7. Integration Techniques, lHpitals o Rule, and Improper Integrals 7.7 Improper Integrals Note. In this section, we are interested in nding the area under a curve over an innite interval. This arises, in particular, in probability and statistics when looking at, for example, the area under the normal distribution. Denition. Integrals with innite limits of integration are improper integrals : 1. If f (x) is continuous on [a, ), then b f (x) dx = lim a b f (x) dx. a 2. If f (x) is continuous on (, b], then b b f (x) dx = lim a f (x) dx. a 3. If f (x) is continuous on (, ), then c f (x) dx = f (x) dx + c f (x) dx. 7.7 Improper Integrals 2 In parts 1 and 2, if the limit is nite, the improper integral converges and the limit is the value of the improper integral. If the limit fails to exist, the improper integral diverges. In part 3, the integral on the left-hand side of the e ...
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c7s7bw
ETSU, MATH 1920
Excerpt: ... 7.7 Improper Integrals 1 Chapter 7. Integration Techniques, lHpitals o Rule, and Improper Integrals 7.7 Improper Integrals Note. In this section, we are interested in nding the area under a curve over an innite interval. This arises, in particular, in probability and statistics when looking at, for example, the area under the normal distribution. Denition. Integrals with innite limits of integration are improper integrals : 1. If f (x) is continuous on [a, ), then b f (x) dx = lim a b f (x) dx. a 2. If f (x) is continuous on (, b], then b b f (x) dx = lim a f (x) dx. a 3. If f (x) is continuous on (, ), then c f (x) dx = f (x) dx + c f (x) dx. 7.7 Improper Integrals 2 In parts 1 and 2, if the limit is nite, the improper integral converges and the limit is the value of the improper integral. If the limit fails to exist, the improper integral diverges. In part 3, the integral on the left-hand side of t ...
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c7s7
ETSU, MATH 1920
Excerpt: ... 7.7 Improper Integrals 1 Chapter 7. Integration Techniques, lHpitals o Rule, and Improper Integrals 7.7 Improper Integrals Note. In this section, we are interested in nding the area under a curve over an innite interval. This arises, in particular, in probability and statistics when looking at, for example, the area under the normal distribution. Denition. Integrals with innite limits of integration are improper integrals : 1. If f (x) is continuous on [a, ), then b f (x) dx = lim a b f (x) dx. a 2. If f (x) is continuous on (, b], then b b f (x) dx = lim a f (x) dx. a 3. If f (x) is continuous on (, ), then c f (x) dx = f (x) dx + c f (x) dx. 7.7 Improper Integrals 2 In parts 1 and 2, if the limit is nite, the improper integral converges and the limit is the value of the improper integral. If the limit fails to exist, the improper integral diverges. In part 3, the integral on the left-hand side of t ...
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10_2
Morningside, MATH 206
Excerpt: ... 10.2 Lecture Notes Math 206 09/15/08 Mammenga DETECTING CONVERGENCE, ESTIMATING LIMITS A. Sometimes it is difficult to determine whether an improper integral converges using the limit technique described in the previous section. The following two tests may help in such situations. B. The Comparison Test (for nonnegative improper integrals ): Let f and g be continuous functions. Suppose that for all x a, 0 f (x) g(x). Then, if a g(x) dx converges, so does a f (x) dx, and g(x) dx. a if a f (x) dx diverges, so does C. The Absolute Comparison Test (for improper integrals ): Suppose that |f (x)| dx converges. Then a a f (x) dx also converges, and |f (x)| dx. a f (x) dx a TEAM ACTIVITIES: 1. Suppose that 1 < f (x) < g(x) for all x 0. (a) Rank the three values 1, 1 1 , and in increasing order. f (x) g(x) 1 1 r , and (f (x) (g(x)r 1 , and (f (x)r (b) Suppose that r 1. Rank the three values 1, in increasing order. (c) Suppose that 0 < r < 1. Rank the three values 1, 1 in increasing order. ...
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c8s8
ETSU, MATH 1920
Excerpt: ... 8.8 Improper Integrals 1 Chapter 8. Techniques of Integration 8.8 Improper Integrals Note. In this section, we are interested in nding the area under a curve over an innite interval. This arises, in particular, in probability and statistics when looking at, for example, the area under the normal distribution. Denition. Integrals with innite limits of integration are improper integrals ot Type I: 1. If f (x) is continuous on [a, ), then a b f (x) dx = lim b f (x) dx. a 2. If f (x) is continuous on (, b], then b b f (x) dx = lim a f (x) dx. a 3. If f (x) is continuous on (, ), then c f (x) dx = f (x) dx + c f (x) dx. 8.8 Improper Integrals 2 In parts 1 and 2, if the limit is nite, the improper integral converges and the limit is the value of the improper integral. If the limit fails to exist, the improper integral diverges. In part 3, the integral on the left-hand side of the equation converges if both improper ...
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c8s8
ETSU, MATH 1920
Excerpt: ... 8.8 Improper Integrals 1 Chapter 8. Techniques of Integration 8.8 Improper Integrals Note. In this section, we are interested in nding the area under a curve over an innite interval. This arises, in particular, in probability and statistics when looking at, for example, the area under the normal distribution. Denition. Integrals with innite limits of integration are improper integrals ot Type I: 1. If f (x) is continuous on [a, ), then b f (x) dx = lim a b f (x) dx. a 2. If f (x) is continuous on (, b], then b b f (x) dx = lim a f (x) dx. a 3. If f (x) is continuous on (, ), then c f (x) dx = f (x) dx + c f (x) dx. 8.8 Improper Integrals 2 In parts 1 and 2, if the limit is nite, the improper integral converges and the limit is the value of the improper integral. If the limit fails to exist, the improper integral diverges. In part 3, the integral on the left-hand side of the equation converges if both improper i ...
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day25
U. Houston, MATH 1432
Excerpt: ... 1 Chapter 10. SEQUENCES; INDETERMINATE FORMS; IMPROPER INTEGRALS 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 ...
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lecture20_handout
U. Houston, MATH 1432
Excerpt: ... Lecture 20Section 10.7 Improper Integrals Jiwen He 1 Improper Integrals 1 b b What are Improper Integrals ? 1 dx =?, x2 b 1 0 1 dx =? x2 Known: 1 1 1 1 dx = - = - , 0 < a < b, x2 x a a b a a the interval of integration [a, b], 0 < a < b, is bounded, f (x) dx = the function being integrated f (x) = 1 x2 is bounded over [a, b]. By a limit process, we can extend the integration process to unbounded intervals (e.g., [1, ): 1 1 1 dx = lim b x2 1 x2 b 1 1 dx = lim b x2 1 1 1 - 1 b = unbounded functions (e.g., as x 0+ , f (x) = 1 a0+ ): 0 1 dx = x2 lim a 1 dx = lim+ x2 a0 1 1 - a 1 = 1.1 Integrals Over Unbounded Intervals Integrals Over Unbounded Intervals 1 b Let f be continuous on [a, ). We define[0.5ex] a f (x) dx = lim b f (x) dx a The improper integral converges if the limit exists. The improper integral diverges if the limit doesn't exist. If lim f (x) = 0, then b a f (x) dx diverges. b b If f is continuous on (-, b], - 0 f (x) dx = lim a- f (x) dx. If f ...
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week5
Allan Hancock College, MATH 11164
Excerpt: ... Brief Study Guide Week 5 Integrals Section 5.9 Approximate Integration Key point. Midpoint Rule (page 412). Key point. Trapezoidal Rule (page 413). Consider Example 1 (pages 413 - 415). Key point. Error Bounds (page 415). Watch and Learn from the Video Lessons: Example 2. The Trapezoidal Rule. Example 3. The Midpoint Rule. Key point. Simpson's Rule (page 418). Consider Example 4 (page 418). Watch and Learn from the Video Lessons: Example 5. Approximating Net Change. Key point. Error Bound for Simpson's Rule (page 419). Consider Example 6 (page 420). Consider Example 7 (page 420). Tutorial and self-assessment questions Do Exercise 9, page 421. Section 5.10 Improper Integrals Key point. Definition of an Improper Integral of Type 1 (page 424). Watch and Learn from the Video Lessons: Example 1. An Improper Integral of Type 1. Consider Example 2 (page 425). Consider Example 4 (page 426). Key point. Definition of an Improper Integral of Type 2 (page 427). Consi ...
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206_Exam3_StudyGuide
University of Hawaii - Hilo, MATH 206
Excerpt: ... Exam 3 Study Guide Dr. Brian Wissman Math 206-001 Spring 2009 The second exam will roughly cover sections 7.7, 8.1, 8.2, and 8.3. More specifically, to do well on the exam you should at least know: Improper Integrals , Finite or Infinite Limits of Integration Sequences and Limits of Sequences (L`Hospital's Rule) Series, Sequence of Partial Sums and their Relationship Determine if a Geometric Series Converges, and find its sum Determine if a Telescoping Series Converges, and find its sum N th Term Divergence Test Integral Test P-Series Note: The basic trigonometric identities will be given to you, but you will need to know when and how to apply them. You are also expected to know how to identify and use the techniques of integration from Chapter 5 and 6. (Such as the u-substitution, by parts, and basic anti-derivatives.) ...
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lecturenotes17
N. Illinois, M 230
Excerpt: ... Lecture notes 8.8: Improper Integrals b In dening a denite integral a f (x)dx we dealt with a function f de- ned on a nite interval [a, b] and we assumed that f does not have an innite discontinuity. Now we extend the concept of a denite integral to the case where the interval is innite and also to the case where f has an innite discontinuity in [a, b]. In either case, the integrals are called indenite integrals. The major application of these is in probability distributions. b Denition of an improper integral of type 1: If number t a, then t a f (x)dx exists for every f (x)dx = lim a b t a f (x)dx provided this limit exists (as a nite number). If t f (x)dx exists for every number t b, then b b f (x)dx = lim t t f (x)dx b provided this limit exists (as a nite number). The improper integrals a b f (x)dx and f (x)dx are called convergent if the corresponding limit exists and divergent if the limit does not ...
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notes Overview of Improper Integrals
Princeton, MATH 104
Textbook:
Thomas' Calculus: Early Transcendentals
Excerpt: ... Overview of Improper Integrals MAT 104 Frank Swenton, Summer 2000 Definitions A proper integral is a definite integral where the interval is finite and the integrand is defined and continuous at all points in the interval. Proper integrals always converge, that is, always give a finite area The trouble spots of a definite integral are the points in the interval of integration that make it an improper integral, i.e., keep it from being proper. They are of two types: a. Points where the integrand is undefined or discontinuous b. and - are always trouble spots when they appear as limits of integration A simple improper integral is an improper integral with only one trouble spot, that trouble spot being at an endpoint of the interval. Simple improper integrals are defined to be the appropriate limits of proper integrals, e.g.: 1 1 1 1 dx = lim dx 0+ x 0 x If the limit exists as a real number, then the simple improper integral is called convergent. If the limit doesn't exist as a real number, the s ...
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exsheet6
Texas A&M, MATH 152
Excerpt: ... Sivakumar Example Sheet 6 Innite and Improper Integrals MATH 152H Material presented here is extracted from Stewarts text as well as from R. G. Bartles The elements of real analysis. Innite Integrals: These integrals are also called improper integrals of the rst kind; we shall, however, use the term innite integrals, which is due to the great British analyst G. H. Hardy (18771947). Suppose a is a xed real number, and that f is Riemann integrable on the interval [a, c] for every c > a. The innite integral of f over [a, ) is dened as follows: c f (x) dx := lim a c f (x) dx, a provided the limit exists (as a nite number). We shall say that the innite integral does not exist if the aforesaid limit does not exist. a f (x) dx 1. Find all values of the real number for which the innite integral exist. 2. Find all values of the real number for which the innite integral exist. 3. Find all values of the real number for whic ...
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