MAA 4212 Improper Integrals


 Kristopher Byrd
 5 years ago
 Views:
Transcription
1 Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly welldefined, is too restrictive for mny purposes; there re functions which we intuitively feel ought to be integrble, but which re not Riemnn integrble ccording to the definition. For exmple, the expression t dt mkes no sense s Riemnn integrl, since the integrnd is not defined t t = 0. Even if we fix tht problem, by defining function tht s t 1/2 for t > 0 nd (sy) 0 for t = 0, this new function is still not Riemnnintegrble over [0, 1] becuse it isn t bounded. However, if formlly mke the chnge of vribles t = u 2 ( formlly mens shoot first, sk questions bout vlidity lter ), the integrl bove gets trnsformed into du, which is s nice n integrl s they come. Furthermore, if we go bck to our originl integrl nd think of it s representing re under curve, there is useful sense in which this re is finite: tke the re below the curve between t = ɛ nd t = 1, nd let ɛ 0. Either wy of looking t the originl integrl, the nswer we formlly clculte is 2. These considertions suggest tht we ought to enlrge the clss of functions we re willing to cll integrble, nd modify our definition of integrl. The types of integrls we ll del with in this hndout re often clled improper integrls, but we ll simply cll them integrls here. Terminology. In this hndout, the words integrl nd integrble will not be synonymous with Riemnn integrl nd Riemnnintegrble. (In Rosenlicht, they re synonymous, but we will need to be clerer here on wht notion of integrtion we re tlking bout.) We will use nottion Riemnn f(t)dt (with the word Riemnn in front of the integrl sign) to denote the Riemnn integrl. Whenever we write hypotheses such s Let f : [, b] R, we understnd this s shorthnd for Let, b R with < b nd let f : [, b] R; nlogous interprettions pply if [, b] is replced by (, b], [, b), or (, b). Also, we write lim x nd lim x in plce of lim x, lim x + respectively. 1 Integrls over bounded intervls. Definition 1. We will sy tht relvlued function f is GRintegrble (for generlized Riemnn integrble ) on the intervl [, b] if either (i) f is defined on (, b] nd is Riemnnintegrble over [y, b] for ll y (, b], nd lim y Riemnn y f(t)dt exists; or 1
2 (ii) f is defined on [, b) nd is Riemnnintegrble over [, y] for ll y [, b), nd lim y b Riemnn y f(t)dt exists. (This is temporry definition tht will be generlized nd finlized in Definition 3.) Note tht if f stisfies both conditions (i) nd (ii), then it is Riemnnintegrble over [, b]. In prticulr, every Riemnnintegrble function is GRintegrble. Note. The terms generlized Riemnn integrl nd GRintegrble re specific to these notes; they re not stndrd terminology. Exercise. 1. Suppose f : [, b] R is Riemnnintegrble on [, b]. Prove tht for ll c [, b], the functions g, h defined by g(x) = Riemnn x c f(t)dt, h(x) = Riemnn c x f(t) dt re continuous. In prticulr, Exercise 1 implies tht if f is Riemnnintegrble on [, b], then lim [Riemnn y y f(t)dt] = Riemnn f(t)dt (1) nd lim [Riemnn y b y f(t)dt] = Riemnn f(t)dt. (2) This suggests using equtions (1) nd (2) to define the integrl in certin nonriemnnintegrble cses. Definition 2. Let f : [, b] R. If f stisfies condition (i) in Definition 1, we define the generlized Riemnn integrl f(t)dt = lim y [Riemnn Similrly if f stisfies condition (ii) in Definition 1, we define f(t)dt = lim y b [Riemnn In both cses we will sy tht f is GRintegrble on [, b]. y y f(t)dt]. (3) f(t)dt]. (4) Note. In plce of sying f is GRintegrble, we often sy the integrl exists or the integrl converges. Equtions (1) (2) show tht there is no mbiguity in Definition 2; if f stisfies both (i) nd (ii) in Definition 1, then the limits in equtions (3) nd (4) re equl. Moreover, if f is Riemnnintegrble on [c, d] then d c f(t)dt = Riemnn d c f(t)dt. Hence if f is GRintegrble on [, b] we my write equtions (3) nd (4) more simply s 2
3 f(t)dt = lim y y y f(t)dt = lim y b f(t)dt. (5) In the exercises below we will develop useful comprison test for telling whether certin functions re GRintegrble. To prove the comprison test vlid it is helpful to hve the following simple lemm, which is n nlog of (nd is equivlent to) the Cuchy criterion for sequences. (This is essentilly the Proposition on p. 74 of Rosenlicht, but it ws phrsed there in too nrrow wy for our purposes.) Lemm 1. Let (E, d), (E, d ) be metric spces. Assume tht p 0 is cluster point of E, tht E is complete, nd tht f is function from E to E. Then lim p p0 f(p) exists iff for ll ɛ > 0 there exists δ > 0 such tht d (f(p), f(q)) < ɛ whenever d(p, p 0 ) nd d(q, p 0 ) re both < δ. Proof. ( ) Assume the limit exists nd hs vlue L E. Let ɛ > 0. Then there exists δ such tht d(p, p 0 ) < δ implies d (f(p), L) < ɛ/2. Hence if p, q B δ (p 0 ), the tringle inequlity implies d (f(p), f(q)) < ɛ. ( ) Let ɛ > 0, nd choose δ > 0 such tht p, q B δ (p 0 ) implies d (f(p), f(q)) < ɛ/2. Since p 0 is cluster point of E there exists sequence {p n } converging to p 0. Let N be such tht n N implies p n B δ (p 0 ). Then for n, m N we hve d (f(p n ), f(p m )) < ɛ/2, so the sequence {f(p n )} is Cuchy. Since E is complete, this sequence converges, sy to L. Let n N be such tht d (f(p n ), L) < ɛ/2 nd let q B δ (p 0 ). Then d (f(q), L) d (f(q), f(p n )) + d (f(p n ), L) < ɛ, so lim q p0 f(q) exists (nd equls L). An equivlent form of this lemm is the first sentence of: Lemm 1. Let the hypotheses be s in Lemm 1. Then lim p p0 f(p) exists iff for every sequence {p n } tht converges to p 0, the sequence f(p n ) converges. If lim p p0 f(p) exists, then it equls lim n f(p n ) for every sequence {p n } tht converges to p 0. Remrk. We hve used weker version of this lemm before: lim p p0 f(p) exists iff there exists L such tht for every sequence {p n } tht converges to p 0, the sequence f(p n ) converges to L. Lemm 1 strengthens the impliction by removing the ssumption tht the sequences {f(p n )} hve the sme limit. Proof of Lemm 1. ( direction of first sentence of conclusion): Turn the crnk. ( direction of first sentence of conclusion, nd second sentence of conclusion) Suppose tht whenever p n p 0, {f(p n )} converges. Choose two such sequences {p n }, {q n } nd suppose f(p n ) L 1 while f(q n ) L 2. The spliced sequence p 1, q 1, p 2, q 2,... lso converges to p 0, but if L 1 L 2, the sequence f(p 1 ), f(q 1 ), f(p 2 ), f(q 2 ),... cnnot converge. Hence L 1 = L 2 ; i.e for ll sequences {p n } converging to p 0, the limiting vlue of f(p n ) is the sme, sy L. Now suppose tht lim p p0 f(p) L or doesn t exist. Then there exists 3
4 ɛ > 0 such tht for ll n, there exists p n B 1/n (p 0 ) such tht d (f(p n ), L) ɛ. Since p n p 0, this is contrdiction. Exercises. 2. Let b >. Prove tht f : x 1 is GRintegrble on [, b] iff p < 1. (Here p cn (x ) p be ny rel number. Thus for 0 < p < 1, f is GRintegrble but not Riemnnintegrble on [, b].) 3. Let b >. Assume f is Riemnnintegrble on [y, b] whenever < y b. (Note tht ny function continuous on (, b] stisfies this criterion, even if not defined or continuous t.) Assume there exists function g : (, b] R, GRintegrble on [, b], such tht f(x) g(x), x (, b]. Prove tht f is GRintegrble on [, b] nd tht f(x)dx g(x)dx. 4. Let b >. Assume f is continuous on (, b] nd tht, for some p < 1, the function x (x ) p f(x) is bounded on (, b]. Prove tht f is GRintegrble on [, b]. 5. Stte the nlogs of exercises 13 with the roles of nd b reversed (i.e. with (, b] replced by [, b)). Essentilly the sme proofs work, of course. 6. Without using ny reference to trigonometric functions or their inverses, prove tht t 2 dt exists (i.e. tht (1 t 2 ) 1/2 is GRintegrble on [0, 1]). Remrk: The vlue of this integrl cn be tken s the definition of π/2. In this hndout, we will refer to point x 0 R s singulrity of f if x 0 is in the closure of the domin of f, but there exists no closed intervl contining x 0 over which f, extended rbitrrily to x 0 if x 0 / domin(f), is Riemnnintegrble. To mke the concept more concrete, it s useful to picture function which is defined nd continuous ner x 0 but not t x 0, nd for which lim x x0 f(x) does not exist (e.g. 0 is singulrity of x 1/x). Grdenvriety functions to which Definitions 1 nd 2 pply re functions tht re continuous on the interior of n intervl but hve singulrity t one endpoint or the other, but not both. We next wnt to extend our definition of GRintegrble on [, b] to include functions tht hve singulrities t more thn one point (e.g. both endpoints) nd/or t n interior point of the intervl of integrtion. Our extension will be bsed on the next exercise, which you should think of s generliztion of the Proposition t the bottom of p. 123 in Rosenlicht. Exercise. 7. Prove tht f is GRintegrble over [, b], in the sense of Definition 2, if nd only if t lest one of the following two conditions holds: (i) for ll c (, b), f is GRintegrble over (, c] nd Riemnnintegrble over [c, b], or (ii) for ll c (, b), f is Riemnnintegrble over [, c] nd GRintegrble over [c, b). When the integrls exist, prove tht 4
5 (End of exercise 7.) f(t)dt = c f(t)dt + c f(t)dt. (6) Now suppose we hve function f defined on [, b], continuous except for n interior singulrity t single point c (, b). If the GRintegrls over [, c] nd [c, b] exist, we cn define the GR integrl of f over [, b] by tking eqution (6) s definition. Similrly, if f is continuous on the interior but singulr t both endpoints, nd if for some interior point c the GR integrls over [, c] nd [c, b] exist, then we cn gin tke (6) s definition of the lefthnd side. Finlly, if we hve function which is continuous [, b] except for singulr points s 1,..., s n, we cn chop up [, b] into finite number of subintervls on which f hs only one singulrity (intersperse nonsingulr points y i with the s i s), nd use the nlog of eqution (6) with one term for ech subintervl to define the lefthnd side. Looking over wht we ve just sid, we see tht we never relly needed f to be continuous off the set of singulr points (though tht s most commonly wht we see in prctice). Our forml definition becomes: Definition 3. Suppose the relvlued function f hs the following property: there exist points s 0, s 1,..., s n+1, with = s 0 nd b = s n+1, such tht f is defined t every point of [, b] except possibly the s i s, nd such tht for 0 i n, f is GRintegrble over [s i, s i+1 ] in the sense of Definition 1. Then we sy tht f is GRintegrble over [, b], nd define n si+1 f(t)dt = f(t)dt. (7) s i i=0 (Note tht we don t require f to be singulr t the s i s; we simply llow it. In generl, to pply Definition 1 we ll hve to intersperse nonsingulr points between the singulr points.) There is potentil problem with this definition. In generl, if f is GRintegrble over [, b], there will be infinitely mny choices for the s i. For exmple, if f(x) = [x(1 x)] 1/2 on [0,1], we could choose s 1 to be ny number strictly between 0 nd 1. For the integrl over [0, 1] to be welldefined, we need to know tht the righthnd side of (6) does not depend on where we put the nonsingulr s i s. Exercise. 8. Prove tht for functions GRintegrble ccording to Definition 3, f(t)dt is welldefined (i.e. does not depend on the choice of the points s i ). 2 Integrls over unbounded intervls. Next we wnt to llow for the possibility of integrting functions over infinite intervls (e.g. [0, )). The most intuitive wy to do this is the following. Definition 4. Let R. We sy f is GRintegrble on [, ) (or f(t)dt 5
6 exists, or f(t)dt converges ) iff (i) for every y >, f is GRintegrble over [, y] (in the sense of Definition 3), nd (ii) lim y y f(t)dt exists. In the GRintegrble cse, we define f(t)dt to be the vlue of this limit. We define GRintegrble over (, ] nd f(t)dt similrly. We sy f is GRintegrble on (, ) if it is GRintegrble on (, 0] nd [0, ), in which cse we define f(t)dt = 0 f(t)dt + 0 f(t)dt. Exercises. 9. Prove tht, in the definition of integrbility over (, ), the number 0 could hve been replced by ny rel number without chnging the set of functions being clled GRintegrble or (in the GRintegrble cse) the vlue of the integrl. 10. Let > 0. Prove tht 11. Determine ll vlues of p for which 0 1 dx exists iff p > 1. (Here p cn be ny rel number.) x p 1 dx exists. x p 12. Let R. Assume f is GRintegrble on [, y] for ll y >. Assume there exists function g : (, ) R, GRintegrble on [, ), such tht f(x) g(x), x >. Prove tht f is GRintegrble on [, ) nd tht f(x)dx g(x)dx. 13. Let R. Assume f is continuous on (, ) nd tht, for some p > 1 nd some q < 1, the function x ((x ) p + (x ) q ) f(x) is bounded on (, ). Prove tht f(t)dt exists. 14. Let ɛ > 0 nd let r > 1. Prove tht, no mtter how smll ɛ is or how lrge r is, 0 x r e ɛx dx converges. 15. Suppose f is defined on [, ) nd f(x) 0 x. Prove tht if f(x)dx exists, then lim inf x f(x) = 0. (Note: previously we defined lim inf only for sequences, but you should be ble to figure out how to extend the definition to the current sitution.) Remrk. The sttement in exercise 15 would be flse if lim inf were replced by lim. First, lim x f(x) doesn t hve to exist for f(x)dx to exist. Second, it is even possible for the integrl to converge if there is sequence x n for which f(x n ). As n exmple, consider function f which is zero most plces, except for tringulr spikes centered t the positive integers. For the spike centered t n, let the bse of the spike hve width 2 2n nd height 2 2 n, so tht the tringle hs re 2 n. Then it s not hrd to show tht f is GRintegrble on [0, ) nd tht the integrl equls the (convergent) geometric series 1 2 n, even though f(n). If we drop the restriction tht f be nonnegtive, it is esy to come up with other exmples of counterintuitive phenomen; see exercises ChngeofVribles Formul. Often forml chnge of vribles mde to simplify the computtion of n integrl turns proper integrl into n improper one, or vicevers. Sometimes chnge of vribles turns one improper integrl into nother. One needs to know whether such chnges of vrible re vlid. For simplicity, we will stte the theorem only for functions 6
7 defined on n intervl of the form (, b] or [b, ) nd which stisfy the corresponding prts of Definitions 1 or 4. A more generl sttement isn t hrd to prove, but is messier to stte. ChngeofVribles Theorem. Suppose f is continuous on n intervl I, where either (i) I = (, b] or (ii) I = [b, ). Let φ : I R be continuous on I nd continuously differentible on the interior of I. In cse (i), suppose tht lim t φ(t) = c, where we llow the symbol c to stnd for rel number or for. (Thus we ssume tht either the limit exists, or tht lim t φ(t) =.) Similrly, in cse (ii), suppose tht lim φ(t) = c, t where gin c my stnd for. Then in ech cse ((i) nd (ii)) either both of the integrls f(φ(t))φ (t)dt, c f(x)dx (8) exist, or neither does. (If c = or c φ(b), see the lst remrk t the end of these notes.) When the integrls exist, they re equl. Proof. We will write out the proof only for the cse in which I = (, b] nd c < φ(b) is rel number; the other cses re similr. Assume the second integrl in (8) exists. Note tht for u >, the hypotheses of Corollry 3 on p. 128 of Rosenlicht (the chngeofvribles formul for proper integrls) re stisfied with U = (u, b). By hypothesis lim φ(b) y c y f(x)dx exists nd lim u φ(u) = c. Hence c f(x)dx = lim y c y f(x)dx = lim u φ(u) f(x)dx = lim f(φ(t))φ (t)dt. u u Thus the limit on the extreme right, which is the definition of f(φ(t))φ (t)dt, exists nd equls the integrl on the extreme left. Conversely, suppose tht the first integrl in (8) exists. Then, s bove, we hve f(φ(t))φ (t)dt = lim u u f(φ(t))φ (t)dt = lim f(x)dx. u φ(u) Let y n c. Since lim t φ(t) = c, the Intermedite Vlue Theorem implies tht φ mps the intervl (, + δ] onto (c, φ( + δ)] for ll δ < b. Thus there exists sequence u n such tht y n = φ(u n ) for ll n. Continuing the chin of equlities bove, we then hve lim f(x)dx = lim f(x)dx = lim f(x)dx. u φ(u) n φ(u n) n y n 7
8 Applying Lemm 1, lim y c y f(x)dx exists nd equls the first integrl in (8). Exercises. 16. Redo exercise 10 by using exercise 1 nd chnge of vribles. 17. Prove tht if f : [0, ) R is nonnegtive, decresing function, with lim x f(x) = 0, then 0 f(x) sin(x)dx converges. 18. Prove tht 0 sin(x)dx does not converge, but tht 0 sin(x 2 ) dx nd 0 x sin(x 2 )dx do converge. (To understnd wht is enbling the lst two integrls to converge, even though the integrnd is not pproching 0 nd, in the lst integrl, is even unbounded it is instructive to grph the integrnds.) 19. Prove tht 1 0 sin( 1 )dx exists. x Finl remrks. 1. From Definition 3 nd eqution (6) it is cler tht f GRintegrble on [, b] does not require the function f even to be defined t nd b. Even if f() nd f(b) re defined, the vlues f(), f(b) ffect neither GRintegrbility (or Riemnn integrbility for tht mtter) nor the vlue of the integrl. Therefore we define GRintegrble on (, b], GRintegrble on [, b), nd GRintegrble on (, b) ll to men the sme thing s GRintegrble on [, b]. The terminology GRintegrble on (, b) is the most flexible, since it llows for the cses = nd b =. 2. So fr we hve defined integrls f(x) dx only when < b. We extend our definition to llow for b in the sme wy s for Riemnn integrls: In the generlized cse if > b we sy tht f(x) dx exists iff f is GRintegrble on (b, ), in which cse we define f(x) dx = b f(x) dx. (We hve used openintervl nottion here in order include the cses in which or b is infinite.) For R we declre every function to be GRintegrble over the intervl [, ], with f(x) dx = 0. 8
Advanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More informationOverview of Calculus I
Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More information5.7 Improper Integrals
458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the
More informationUNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3
UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More informationn f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1
The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the
More informationf(x)dx . Show that there 1, 0 < x 1 does not exist a differentiable function g : [ 1, 1] R such that g (x) = f(x) for all
3 Definite Integrl 3.1 Introduction In school one comes cross the definition of the integrl of rel vlued function defined on closed nd bounded intervl [, b] between the limits nd b, i.e., f(x)dx s the
More informationChapter 0. What is the Lebesgue integral about?
Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous
More informationRiemann is the Mann! (But Lebesgue may besgue to differ.)
Riemnn is the Mnn! (But Lebesgue my besgue to differ.) Leo Livshits My 2, 2008 1 For finite intervls in R We hve seen in clss tht every continuous function f : [, b] R hs the property tht for every ɛ >
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums  1 Riemnn
More informationThe First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).
The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples
More informationImproper Integrals. Type I Improper Integrals How do we evaluate an integral such as
Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph
More informationExam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH1105 Instructor: Attila Máté 1
Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixedpoint itertion to converge when solving the eqution
More informationLecture 3 ( ) (translated and slightly adapted from lecture notes by Martin Klazar)
Lecture 3 (5.3.2018) (trnslted nd slightly dpted from lecture notes by Mrtin Klzr) Riemnn integrl Now we define precisely the concept of the re, in prticulr, the re of figure U(, b, f) under the grph of
More informationHomework 4. (1) If f R[a, b], show that f 3 R[a, b]. If f + (x) = max{f(x), 0}, is f + R[a, b]? Justify your answer.
Homework 4 (1) If f R[, b], show tht f 3 R[, b]. If f + (x) = mx{f(x), 0}, is f + R[, b]? Justify your nswer. (2) Let f be continuous function on [, b] tht is strictly positive except finitely mny points
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More informationAdvanced Calculus: MATH 410 Uniform Convergence of Functions Professor David Levermore 11 December 2015
Advnced Clculus: MATH 410 Uniform Convergence of Functions Professor Dvid Levermore 11 December 2015 12. Sequences of Functions We now explore two notions of wht it mens for sequence of functions {f n
More information7.2 Riemann Integrable Functions
7.2 Riemnn Integrble Functions Theorem 1. If f : [, b] R is step function, then f R[, b]. Theorem 2. If f : [, b] R is continuous on [, b], then f R[, b]. Theorem 3. If f : [, b] R is bounded nd continuous
More informationMain topics for the First Midterm
Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 23, Sections 4.14.8, nd Sections 5.15.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the
More informationTHE EXISTENCEUNIQUENESS THEOREM FOR FIRSTORDER DIFFERENTIAL EQUATIONS.
THE EXISTENCEUNIQUENESS THEOREM FOR FIRSTORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrdlindeloftheorem/ This document is proof of the existenceuniqueness theorem
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 200910 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationNotes on length and conformal metrics
Notes on length nd conforml metrics We recll how to mesure the Eucliden distnce of n rc in the plne. Let α : [, b] R 2 be smooth (C ) rc. Tht is α(t) (x(t), y(t)) where x(t) nd y(t) re smooth rel vlued
More informationImproper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:
Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl
More informationThe HenstockKurzweil integral
fculteit Wiskunde en Ntuurwetenschppen The HenstockKurzweil integrl Bchelorthesis Mthemtics June 2014 Student: E. vn Dijk First supervisor: Dr. A.E. Sterk Second supervisor: Prof. dr. A. vn der Schft
More informationf(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral
Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one
More informationUNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE
UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE 1. Pointwise Convergence of Sequence Let E be set nd Y be metric spce. Consider functions f n : E Y for n = 1, 2,.... We sy tht the sequence
More informationReview of Riemann Integral
1 Review of Riemnn Integrl In this chpter we review the definition of Riemnn integrl of bounded function f : [, b] R, nd point out its limittions so s to be convinced of the necessity of more generl integrl.
More informationSYDE 112, LECTURES 3 & 4: The Fundamental Theorem of Calculus
SYDE 112, LECTURES & 4: The Fundmentl Theorem of Clculus So fr we hve introduced two new concepts in this course: ntidifferentition nd Riemnn sums. It turns out tht these quntities re relted, but it is
More informationx = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b
CHAPTER 5. INTEGRALS 61 where nd x = b n x i = 1 (x i 1 + x i ) = midpoint of [x i 1, x i ]. Problem 168 (Exercise 1, pge 377). Use the Midpoint Rule with the n = 4 to pproximte 5 1 x e x dx. Some quick
More informationThe final exam will take place on Friday May 11th from 8am 11am in Evans room 60.
Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23
More informationA BRIEF INTRODUCTION TO UNIFORM CONVERGENCE. In the study of Fourier series, several questions arise naturally, such as: c n e int
A BRIEF INTRODUCTION TO UNIFORM CONVERGENCE HANS RINGSTRÖM. Questions nd exmples In the study of Fourier series, severl questions rise nturlly, such s: () (2) re there conditions on c n, n Z, which ensure
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 7.. Introduction to Integrtion. 7. Integrl Clculus As ws the cse with the chpter on differentil
More informationProperties of the Riemann Integral
Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University Februry 15, 2018 Outline 1 Some Infimum nd Supremum Properties 2
More informationDefinite integral. Mathematics FRDIS MENDELU
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion  re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More information11 An introduction to Riemann Integration
11 An introduction to Riemnn Integrtion The PROOFS of the stndrd lemms nd theorems concerning the Riemnn Integrl re NEB, nd you will not be sked to reproduce proofs of these in full in the exmintion in
More informationRecitation 3: More Applications of the Derivative
Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech
More informationDefinite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion  re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function
More informationAbstract inner product spaces
WEEK 4 Abstrct inner product spces Definition An inner product spce is vector spce V over the rel field R equipped with rule for multiplying vectors, such tht the product of two vectors is sclr, nd the
More informationINTRODUCTION TO INTEGRATION
INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide
More informationThe area under the graph of f and above the xaxis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the xxis etween nd is denoted y f(x) dx nd clled the
More informationAppendix to Notes 8 (a)
Appendix to Notes 8 () 13 Comprison of the Riemnn nd Lebesgue integrls. Recll Let f : [, b] R be bounded. Let D be prtition of [, b] such tht Let D = { = x 0 < x 1
More informationII. Integration and Cauchy s Theorem
MTH6111 Complex Anlysis 200910 Lecture Notes c Shun Bullett QMUL 2009 II. Integrtion nd Cuchy s Theorem 1. Pths nd integrtion Wrning Different uthors hve different definitions for terms like pth nd curve.
More informationW. We shall do so one by one, starting with I 1, and we shall do it greedily, trying
Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)
More informationCalculus III Review Sheet
Clculus III Review Sheet 1 Definitions 1.1 Functions A function is f is incresing on n intervl if x y implies f(x) f(y), nd decresing if x y implies f(x) f(y). It is clled monotonic if it is either incresing
More informationLecture 3. Limits of Functions and Continuity
Lecture 3 Limits of Functions nd Continuity Audrey Terrs April 26, 21 1 Limits of Functions Notes I m skipping the lst section of Chpter 6 of Lng; the section bout open nd closed sets We cn probbly live
More informationCalculus II: Integrations and Series
Clculus II: Integrtions nd Series August 7, 200 Integrls Suppose we hve generl function y = f(x) For simplicity, let f(x) > 0 nd f(x) continuous Denote F (x) = re under the grph of f in the intervl [,x]
More information1 The Riemann Integral
The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationp(t) dt + i 1 re it ireit dt =
Note: This mteril is contined in Kreyszig, Chpter 13. Complex integrtion We will define integrls of complex functions long curves in C. (This is bit similr to [relvlued] line integrls P dx + Q dy in R2.)
More informationLECTURE. INTEGRATION AND ANTIDERIVATIVE.
ANALYSIS FOR HIGH SCHOOL TEACHERS LECTURE. INTEGRATION AND ANTIDERIVATIVE. ROTHSCHILD CAESARIA COURSE, 2015/6 1. Integrtion Historiclly, it ws the problem of computing res nd volumes, tht triggered development
More information38 Riemann sums and existence of the definite integral.
38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the xxis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These
More informationand that at t = 0 the object is at position 5. Find the position of the object at t = 2.
7.2 The Fundmentl Theorem of Clculus 49 re mny, mny problems tht pper much different on the surfce but tht turn out to be the sme s these problems, in the sense tht when we try to pproimte solutions we
More informationIMPORTANT THEOREMS CHEAT SHEET
IMPORTANT THEOREMS CHEAT SHEET BY DOUGLAS DANE Howdy, I m Bronson s dog Dougls. Bronson is still complining bout the textbook so I thought if I kept list of the importnt results for you, he might stop.
More informationLecture 1: Introduction to integration theory and bounded variation
Lecture 1: Introduction to integrtion theory nd bounded vrition Wht is this course bout? Integrtion theory. The first question you might hve is why there is nything you need to lern bout integrtion. You
More informationMath 554 Integration
Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we
More informationThe Riemann Integral
Deprtment of Mthemtics King Sud University 20172018 Tble of contents 1 Antiderivtive Function nd Indefinite Integrls 2 3 4 5 Indefinite Integrls & Antiderivtive Function Definition Let f : I R be function
More informationContinuous Random Variables
STAT/MATH 395 A  PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is relvlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More informationChapter 6. Infinite series
Chpter 6 Infinite series We briefly review this chpter in order to study series of functions in chpter 7. We cover from the beginning to Theorem 6.7 in the text excluding Theorem 6.6 nd Rbbe s test (Theorem
More informationChapter 14. Matrix Representations of Linear Transformations
Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn
More informationIntegrals  Motivation
Integrls  Motivtion When we looked t function s rte of chnge If f(x) is liner, the nswer is esy slope If f(x) is nonliner, we hd to work hrd limits derivtive A relted question is the re under f(x) (but
More informationf(a+h) f(a) x a h 0. This is the rate at which
M408S Concept Inventory smple nswers These questions re openended, nd re intended to cover the min topics tht we lerned in M408S. These re not crnkoutnnswer problems! (There re plenty of those in the
More information4.4 Areas, Integrals and Antiderivatives
. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order
More informationIntegrals along Curves.
Integrls long Curves. 1. Pth integrls. Let : [, b] R n be continuous function nd let be the imge ([, b]) of. We refer to both nd s curve. If we need to distinguish between the two we cll the function the
More informationConvex Sets and Functions
B Convex Sets nd Functions Definition B1 Let L, +, ) be rel liner spce nd let C be subset of L The set C is convex if, for ll x,y C nd ll [, 1], we hve 1 )x+y C In other words, every point on the line
More informationInfinite Geometric Series
Infinite Geometric Series Finite Geometric Series ( finite SUM) Let 0 < r < 1, nd let n be positive integer. Consider the finite sum It turns out there is simple lgebric expression tht is equivlent to
More informationWeek 10: Line Integrals
Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.
More informationpadic Egyptian Fractions
padic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Setup 3 4 pgreedy Algorithm 5 5 pegyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction
More informationMath Advanced Calculus II
Mth 452  Advnced Clculus II Line Integrls nd Green s Theorem The min gol of this chpter is to prove Stoke s theorem, which is the multivrible version of the fundmentl theorem of clculus. We will be focused
More informationMapping the delta function and other Radon measures
Mpping the delt function nd other Rdon mesures Notes for Mth583A, Fll 2008 November 25, 2008 Rdon mesures Consider continuous function f on the rel line with sclr vlues. It is sid to hve bounded support
More informationLine and Surface Integrals: An Intuitive Understanding
Line nd Surfce Integrls: An Intuitive Understnding Joseph Breen Introduction Multivrible clculus is ll bout bstrcting the ides of differentition nd integrtion from the fmilir single vrible cse to tht of
More informationThe Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.
Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F
More informationExample Sheet 6. Infinite and Improper Integrals
Sivkumr Exmple Sheet 6 Infinite nd Improper Integrls MATH 5H Mteril presented here is extrcted from Stewrt s text s well s from R. G. Brtle s The elements of rel nlysis. Infinite Integrls: These integrls
More information4181H Problem Set 11 Selected Solutions. Chapter 19. n(log x) n 1 1 x x dx,
48H Problem Set Selected Solutions Chpter 9 # () Tke f(x) = x n, g (x) = e x, nd use integrtion by prts; this gives reduction formul: x n e x dx = x n e x n x n e x dx. (b) Tke f(x) = (log x) n, g (x)
More informationMath 61CM  Solutions to homework 9
Mth 61CM  Solutions to homework 9 Cédric De Groote November 30 th, 2018 Problem 1: Recll tht the left limit of function f t point c is defined s follows: lim f(x) = l x c if for ny > 0 there exists δ
More informationThe RiemannLebesgue Lemma
Physics 215 Winter 218 The RiemnnLebesgue Lemm The Riemnn Lebesgue Lemm is one of the most importnt results of Fourier nlysis nd symptotic nlysis. It hs mny physics pplictions, especilly in studies of
More information1 Sets Functions and Relations Mathematical Induction Equivalence of Sets and Countability The Real Numbers...
Contents 1 Sets 1 1.1 Functions nd Reltions....................... 3 1.2 Mthemticl Induction....................... 5 1.3 Equivlence of Sets nd Countbility................ 6 1.4 The Rel Numbers..........................
More informationMATH 409 Advanced Calculus I Lecture 22: Improper Riemann integrals.
MATH 409 Advned Clulus I Leture 22: Improper Riemnn integrls. Improper Riemnn integrl If funtion f : [,b] R is integrble on [,b], then the funtion F(x) = x f(t)dt is well defined nd ontinuous on [,b].
More information1 i n x i x i 1. Note that kqk kp k. In addition, if P and Q are partition of [a, b], P Q is finer than both P and Q.
Chpter 6 Integrtion In this chpter we define the integrl. Intuitively, it should be the re under curve. Not surprisingly, fter mny exmples, counter exmples, exceptions, generliztions, the concept of the
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further
More informationc n φ n (x), 0 < x < L, (1) n=1
SECTION : Fourier Series. MATH4. In section 4, we will study method clled Seprtion of Vribles for finding exct solutions to certin clss of prtil differentil equtions (PDEs. To do this, it will be necessry
More informationAnalysis III. Ben Green. Mathematical Institute, Oxford address:
Anlysis III Ben Green Mthemticl Institute, Oxford Emil ddress: ben.green@mths.ox.c.uk 2000 Mthemtics Subject Clssifiction. Primry Contents Prefce 1 Chpter 1. Step functions nd the Riemnn integrl 3 1.1.
More informationFourier series. Preliminary material on inner products. Suppose V is vector space over C and (, )
Fourier series. Preliminry mteril on inner products. Suppose V is vector spce over C nd (, ) is Hermitin inner product on V. This mens, by definition, tht (, ) : V V C nd tht the following four conditions
More informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
More informationHandout: Natural deduction for first order logic
MATH 457 Introduction to Mthemticl Logic Spring 2016 Dr Json Rute Hndout: Nturl deduction for first order logic We will extend our nturl deduction rules for sententil logic to first order logic These notes
More informationdifferent methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s).
Mth 1A with Professor Stnkov Worksheet, Discussion #41; Wednesdy, 12/6/217 GSI nme: Roy Zho Problems 1. Write the integrl 3 dx s limit of Riemnn sums. Write it using 2 intervls using the 1 x different
More informationReview of basic calculus
Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below
More information5.5 The Substitution Rule
5.5 The Substitution Rule Given the usefulness of the Fundmentl Theorem, we wnt some helpful methods for finding ntiderivtives. At the moment, if n ntiderivtive is not esily recognizble, then we re in
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationFUNDAMENTALS OF REAL ANALYSIS by. III.1. Measurable functions. f 1 (
FUNDAMNTALS OF RAL ANALYSIS by Doğn Çömez III. MASURABL FUNCTIONS AND LBSGU INTGRAL III.. Mesurble functions Hving the Lebesgue mesure define, in this chpter, we will identify the collection of functions
More informationCHAPTER 4 MULTIPLE INTEGRALS
CHAPTE 4 MULTIPLE INTEGAL The objects of this chpter re fivefold. They re: (1 Discuss when sclrvlued functions f cn be integrted over closed rectngulr boxes in n ; simply put, f is integrble over iff
More informationChapter 6. Riemann Integral
Introduction to Riemnn integrl Chpter 6. Riemnn Integrl WonKwng Prk Deprtment of Mthemtics, The College of Nturl Sciences Kookmin University Second semester, 2015 1 / 41 Introduction to Riemnn integrl
More informationAntiderivatives/Indefinite Integrals of Basic Functions
Antiderivtives/Indefinite Integrls of Bsic Functions Power Rule: In prticulr, this mens tht x n+ x n n + + C, dx = ln x + C, if n if n = x 0 dx = dx = dx = x + C nd x (lthough you won t use the second
More information