Math 554 Integration


 Ophelia Powers
 1 years ago
 Views:
Transcription
1 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 define the norm of the prtition by P := mx 1 i n x i. where x i := x i x i 1 is the length of the ith subintervl [x i 1, x i ]. Defn. For given prtition P, we define the Riemnn upper sum of function f by U(P, f) := n M i x i where M i denotes the supremum of f over ech of the subintervls [x i 1, x i ]. Similrly, we define the Riemnn lower sum of function f by L(P, f) := n m i x i where m i denotes the infimum of f over ech of the subintervls [x i 1, x i ]. Since m i M i, we note tht L(P, f) U(P, f). for ny prtition P. Defn. Suppose P 1, P 2 re both prtitions of [, b], then P 2 is clled refinement of P 1, denoted by P 1 P 2, if s sets P 1 P 2. Note. If P 1 P 2, it follows tht P 2 P 1 since ech of the subintervls formed by P 2 is contined in subintervl which rises from P 1. Lemm. If P 1 P 2, then nd L(P 1, f) L(P 2, f). U(P 2, f) U(P 1, f).
2 Proof. Suppose first tht P 1 is prtition of [, b] nd tht P 2 is the prtition obtined from P 1 by dding n dditionl point z. The generl cse follows by induction, dding one point t t time. In prticulr, we let nd P 1 := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } P 2 := {x 0 = < x 1 < < x i 1 < z < x i < < b =: x n } for some fixed i. We focus on the upper Riemnn sum for these two prtitions, noting tht the inequlity for the lower sums follows similrly. Observe tht nd U(P 2, f) := i 1 j=1 U(P 1, f) := n M j x j j=1 M j x j + M(z x i 1 ) + M(x i z) + n j=i+1 M j x j where M := sup [xi 1,z] f(x) nd M := sup [z,xi ] f(x). It then follows tht U(P 2, f) U(P 1, f) since M, M M i. Defn. If P 1 nd P 2 re rbitrry prtitions of [, b], then the common refinement of P 1 nd P 2 is the forml union of the two. Corollry. Suppose P 1 nd P 2 re rbitrry prtitions of [, b], then L(P 1, f) U(P 2, f). Proof. Let P be the common refinement of P 1 nd P 2, then L(P 1, f) L(P, f) U(P, f) U(P 2, f). Defn. The lower Riemnn integrl of f over [, b] is defined to be f(x)dx := sup ll prtitions P of [,b] L(P, f). Similrly, the upper Riemnn integrl of f over [, b] is defined to be f(x)dx := inf ll prtitions P of [,b] U(P, f).
3 By the definitions of lest upper bound nd gretest lower bound, it is evident tht for ny function f there holds f(x)dx f(x)dx. Defn. A function f is Riemnn integrble over [, b] if the upper nd lower Riemnn integrls coincide. We denote this common vlue by f(x) dx. Exmples: 1. k dx = k(b ). 2. x dx = 1 2 (b2 2 ). [Hint: Use n i = n(n + 1)/2.] Theorem. A necessry nd sufficient condition for f to be Riemnn integrble is given ɛ > 0, there exists prtition P of [, b] such tht ( ) U(P, f) L(P, f) < ɛ. Proof. First we show tht (*) is sufficient condition. This follows immeditely, since for ech ɛ > 0 tht there is prtition P such tht (*) holds, f(x)dx b f(x)dx U(P, f) L(P, f) < ɛ. Since ɛ > 0 ws rbitrry, then the upper nd lower Riemnn integrls of f must coincide. To prove tht (*) is necessry condition for f to be Riemnn integrble, we let ɛ > 0. By the definition of the upper Riemnn integrl s infimum of upper sums, we cn find prtition P 1 of [, b] such tht Similrly, we hve f(x)dx U(P 1, f) < f(x)dx + ɛ/2 f(x)dx ɛ/2 < L(P 2, f) f(x)dx. Let P be common refinement of P 1 nd P 2, then subtrcting the two previous inequlities implies, U(P, f) L(P, f) U(P 1, f) L(P 2, f) < ɛ. Defn. A Riemnn sum for f for prtition P of n intervl [, b] is defined by R(P, f, ξ) := n f(ξ j ) x j j=1
4 where the ξ j, stisfying x j 1 ξ j x j (1 j n), re rbitrry. Corollry. Suppose tht f is Riemnn integrble on [, b], then there is unique number γ ( = f(x)dx) such tht for every ɛ > 0 there exists prtition P of [, b] such tht if P P 1, P 2, then i.) ii.) iii.) 0 U(P 1, f) γ < ɛ 0 γ L(P 2, f) < ɛ γ R(P 1, f, ξ) < ɛ where R(P 1, f, ξ) is ny Riemnn sum of f for the prtition P 1. In this sense, we cn interpret f(x)dx = lim R(P, f, ξ). P 0 lthough we would ctully need to show little more to be entirely correct. Proof. Since L(P 2, f) γ U(P 1, f) for ll prtitions, we see tht prts i.) nd ii.) follow from the definition of the Riemnn integrl. To see prt iii.), we observe tht m j f(ξ j ) M j nd hence tht But we lso know tht both L(P 1, f) R(P 1, f, ξ) U(P 1, f). L(P 1, f) γ U(P 1, f) nd condition (*) hold, from which prt iii.) follows.
5 Theorem. If f is continuous on [, b], then f is Riemnnintegrble on [, b]. Proof. We use the condition (*) to prove tht f is Riemnnintegrble. If ɛ > 0, we set ɛ 0 := ɛ/(b ). Since f is continuous on [, b], f is uniformly continuous. Hence there is δ > 0 such tht f(y) f(x) < ɛ 0 if y x < δ. Suppose tht P < δ, then it follows tht M i m i ɛ 0 (1 i n). Hence U(P, f) L(P, f) = n (M i m i ) x i ɛ 0 (b ) = ɛ. Theorem. If f is monotone on [, b], then f is Riemnnintegrble on [, b]. Proof. If f is constnt, then we re done. We prove the cse for f monotone incresing. The cse for monotone decresing is similir. We gin use the condition (*) to prove tht f is Riemnnintegrble. If ɛ > 0, we set δ := ɛ/(f(b) f()) nd consider ny prtition P with P < δ. Since f is monotone incresing on [, b], then M i = f(x i ) nd m i = f(x i 1 ). Hence U(P, f) L(P, f) = n = n (M i m i ) x i (f(x i ) f(x i 1 )) x i P n (f(x i ) f(x i 1 )) < δ (f(b) f()) = ɛ. Theorem. (Properties of the Riemnn Integrl) Suppose tht f nd g re Riemnn integrble nd k is rel number, then i.) k f(x) dx = k f(x) dx ii.) f + g dx = f dx + g dx iii.) g f implies g dx f dx. iv.) f dx f dx Proof. To prove prt i.), we observe tht in cse k 0, then sup [xi 1,x i ] kf(x) = km i nd inf [xi 1,x i ] kf(x) = km i. Hence U(P, kf) = ku(p, f) nd L(P, kf) = kl(p, f). In the cse tht k < 0, then sup [xi 1,x i ] kf(x) = km i nd inf [xi 1,x i ] kf(x) = km i. It follows in this cse tht U(P, kf) = kl(p, f) nd L(P, kf) = ku(p, f) nd so k f(x)dx = k b f(x)dx
6 k f(x)dx = k f(x)dx. To prove property ii.) we notice tht sup I (f + g) sup I f + sup I g nd inf I f + inf I g inf I (f + g) for ny intervl I (for exmple, I = [x i 1, x i ]). Hence, (1) L(P, f) + L(P, g) L(P, f + g) U(P, f + g) U(P, f) + U(P, g). Let ɛ > 0, then since f nd g re Riemnn integrble, there exist prtitions P 1, P 2 such tht (2) U(P 1, f) L(P 1, f) < ɛ/2, U(P 2, g) L(P 2, g) < ɛ/2. If we let P be common refinement of P 1 nd P 2, then by combining inequlities (1) nd (2), we see tht see tht U(P, f + g) L(P, f + g) U(P, f) L(P, f) + U(P, g) L(P, g) U(P 1, f) L(P 1, f) + U(P 2, g) L(P 2, g) ɛ/2 + ɛ/2 = ɛ. Property iii.) follows directly from the definition of the upper nd lower integrls using, for exmple, the inequlity sup I g(x) sup I f(x). Property iv.) is proved by pplying property iii.) to the inequlity f f f, from which it follows tht f dx f dx f dx. But this inequlity implies property iv.). Defn. We extend the definition of the integrl to include generl limits of integrtion. These re consistent with our erlier definition. 1. f(x) dx = b f(x) dx = f(x) dx. Theorem. If f is Riemnn integrble on [, b], then it is Riemnn integrble on ech subintervl [c, d] [, b]. Moreover, if c [, b], then (3) f(x) dx = c f(x) dx + b f(x) dx. Proof. We show first tht condition (*) holds for the intervl [c, d]. Suppose ɛ > 0, then by (*) pplied to f over the intervl [, b], we hve tht there exists prtition P of [, b] such tht condition (*) holds. Let P be the refinement obtined from P c
7 which contins the points c nd d. Let P be the prtition obtined by restricting the prtition P to the intervl [c, d], then U(P, f) L(P, f) U( P, f) L( P, f) U(P, f) L(P, f) < ɛ nd so f is Riemnn integrble over [c, d]. To prove the identity (3), we use the fct tht condition (*) holds when f is Riemnn integrble. Let ɛ > 0, then for ɛ/3 > 0, we my pply (*) to ech of the intervls I = [, b], [, c] nd [c, b], respectively, to obtin prtitions P I which stisfy (4) 0 U I (P I, f) I f dx U I(P I, f) L I (P I, f) < ɛ/3. We let P be the prtition of [, b] formed by the union of the two prtitions P [,c], P [c,b], nd P be the common refinement of P nd P [,b]. Observing tht (5) U [,b] ( P, f) = U [,c] ( P 1, f) + U [c,b] ( P 2, f), we cn combine with inequlity (4) to obtin c f dx + c f dx f dx U [,c] ( P, f) c f dx + U [c,b] ( P, f) c f dx < 3ɛ 0 = ɛ. + U [,b] ( P, f) f dx Since ɛ > 0 ws rbitrry, then equlity (3) must hold. Theorem. (Intermedite Vlue Theorem for Integrls) If f is continuous on [, b], then there exists ξ between nd b such tht f(x) dx = f(ξ)(b ). Proof. Since f is continuous on [, b] nd for η := min [,b] f dx b f(x) η mx f(x), [,b] there holds then by the Intermedite Vlue Theorem for continuous functions, there exists ξ [, b] such tht f(ξ) = η. Theorem. (Fundmentl Theorem of Clculus, I. Derivtive of n Integrl) Suppose tht f is continuous on [, b] nd set F (x) := x f(y)dy, then F is differentible nd F (x) = f(x) for < x < b.
8 Proof. Notice tht F (x 0 + h) F (x 0 ) h = x0 +h x 0 f dx h = f(ξ) for some ξ between x 0 nd x 0 + h. Hence, s h 0, then ξ = ξ h converges to x 0 nd so the displyed difference quotient hs limit of f(x 0 ) s h 0. Theorem. (Fundmentl Theorem of Clculus, Prt II. Integrl of Derivtive) Suppose tht F is function with continuous derivtive on [, b], then F (y) dy = F (x) x=b x= := F (b) F (). Proof. Define G(x) := x F (y) dy, nd set H := F G. Since the derivtive of H is identiclly zero by Prt I of the Fundmentl Theorem of Clculus, then the Men Vlue Theorem implies tht H(b) = H(). Expressing this in terms of F nd G gives F (b) F (y) dy = F (), which estblishes the theorem.
7.2 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 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 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 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 informationAdvanced 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 informationCalculus in R. Chapter Di erentiation
Chpter 3 Clculus in R 3.1 Di erentition Definition 3.1. Suppose U R is open. A function f : U! R is di erentible t x 2 U if there exists number m such tht lim y!0 pple f(x + y) f(x) my y =0. If f is di
More informationFor a continuous function f : [a; b]! R we wish to define the Riemann integral
Supplementry Notes for MM509 Topology II 2. The Riemnn Integrl Andrew Swnn For continuous function f : [; b]! R we wish to define the Riemnn integrl R b f (x) dx nd estblish some of its properties. This
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 informationPrinciples of Real Analysis I Fall VI. Riemann Integration
21355 Principles of Rel Anlysis I Fll 2004 A. Definitions VI. Riemnn Integrtion Let, b R with < b be given. By prtition of [, b] we men finite set P [, b] with, b P. The set of ll prtitions of [, b] will
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 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 informationMAT612REAL ANALYSIS RIEMANN STIELTJES INTEGRAL
MAT612REAL ANALYSIS RIEMANN STIELTJES INTEGRAL DR. RITU AGARWAL MALVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR, INDIA302017 Tble of Contents Contents Tble of Contents 1 1. Introduction 1 2. Prtition
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 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 informationPresentation Problems 5
Presenttion Problems 5 21355 A For these problems, ssume ll sets re subsets of R unless otherwise specified. 1. Let P nd Q be prtitions of [, b] such tht P Q. Then U(f, P ) U(f, Q) nd L(f, P ) L(f, Q).
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 informationThe Banach algebra of functions of bounded variation and the pointwise Helly selection theorem
The Bnch lgebr of functions of bounded vrition nd the pointwise Helly selection theorem Jordn Bell jordn.bell@gmil.com Deprtment of Mthemtics, University of Toronto Jnury, 015 1 BV [, b] Let < b. For f
More informationMATH 409 Advanced Calculus I Lecture 18: Darboux sums. The Riemann integral.
MATH 409 Advnced Clculus I Lecture 18: Drboux sums. The Riemnn integrl. Prtitions of n intervl Definition. A prtition of closed bounded intervl [, b] is finite subset P [,b] tht includes the endpoints
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 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 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 informationMath 118: Honours Calculus II Winter, 2003 List of Theorems. Lemma 5.1 (Partition Refinement) If P and Q are partitions of [a, b] such that Q P, then
Mth 118: Honours Clculus II Winter, 2003 List of Theorems Lemm 5.1 (Prtition Refinement) If P nd Q re prtitions of [, b] such tht Q P, then L(P, f) L(Q, f) U(Q, f) U(P, f). Lemm 5.2 (Upper Sums Bound Lower
More informationMATH 409 Advanced Calculus I Lecture 19: Riemann sums. Properties of integrals.
MATH 409 Advnced Clculus I Lecture 19: Riemnn sums. Properties of integrls. Drboux sums Let P = {x 0,x 1,...,x n } be prtition of n intervl [,b], where x 0 = < x 1 < < x n = b. Let f : [,b] R be bounded
More informationMath 118: Honours Calculus II Winter, 2005 List of Theorems. L(P, f) U(Q, f). f exists for each ǫ > 0 there exists a partition P of [a, b] such that
Mth 118: Honours Clculus II Winter, 2005 List of Theorems Lemm 5.1 (Prtition Refinement): If P nd Q re prtitions of [, b] such tht Q P, then L(P, f) L(Q, f) U(Q, f) U(P, f). Lemm 5.2 (Upper Sums Bound
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 informationReview. April 12, Definition 1.2 (Closed Set). A set S is closed if it contains all of its limit points. S := S S
Review April 12, 2017 1 Definitions nd Some Theorems 1.1 Topology Definition 1.1 (Limit Point). Let S R nd x R. Then x is limit point of S if, for ll ɛ > 0, V ɛ (x) = (x ɛ, x + ɛ) contins n element s S
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 informationRiemann Stieltjes Integration  Definition and Existence of Integral
 Definition nd Existence of Integrl Dr. Adity Kushik Directorte of Distnce Eduction Kurukshetr University, Kurukshetr Hryn 136119 Indi. Prtition Riemnn Stieltjes Sums Refinement Definition Given closed
More information1. On some properties of definite integrals. We prove
This short collection of notes is intended to complement the textbook Anlisi Mtemtic 2 by Crl Mdern, published by Città Studi Editore, [M]. We refer to [M] for nottion nd the logicl stremline of the rguments.
More informationChapter 4. Lebesgue Integration
4.2. Lebesgue Integrtion 1 Chpter 4. Lebesgue Integrtion Section 4.2. Lebesgue Integrtion Note. Simple functions ply the sme role to Lebesgue integrls s step functions ply to Riemnn integrtion. Definition.
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 informationReview on Integration (Secs ) Review: Sec Origins of Calculus. Riemann Sums. New functions from old ones.
Mth 20B Integrl Clculus Lecture Review on Integrtion (Secs. 5.  5.3) Remrks on the course. Slide Review: Sec. 5.5.3 Origins of Clculus. Riemnn Sums. New functions from old ones. A mthemticl description
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 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 informationWeek 10: Riemann integral and its properties
Clculus nd Liner Algebr for Biomedicl Engineering Week 10: Riemnn integrl nd its properties H. Führ, Lehrstuhl A für Mthemtik, RWTH Achen, WS 07 Motivtion: Computing flow from flow rtes 1 We observe the
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 informationAdvanced Calculus I (Math 4209) Martin Bohner
Advnced Clculus I (Mth 4209) Spring 2018 Lecture Notes Mrtin Bohner Version from My 4, 2018 Author ddress: Deprtment of Mthemtics nd Sttistics, Missouri University of Science nd Technology, Roll, Missouri
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 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 informationDEFINITE INTEGRALS. f(x)dx exists. Note that, together with the definition of definite integrals, definitions (2) and (3) define b
DEFINITE INTEGRALS JOHN D. MCCARTHY Astrct. These re lecture notes for Sections 5.3 nd 5.4. 1. Section 5.3 Definition 1. f is integrle on [, ] if f(x)dx exists. Definition 2. If f() is defined, then f(x)dx.
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 informationarxiv: v1 [math.ca] 11 Jul 2011
rxiv:1107.1996v1 [mth.ca] 11 Jul 2011 Existence nd computtion of Riemnn Stieltjes integrls through Riemnn integrls July, 2011 Rodrigo López Pouso Deprtmento de Análise Mtemátic Fcultde de Mtemátics, Universidde
More informationProblem Set 4: Solutions Math 201A: Fall 2016
Problem Set 4: s Mth 20A: Fll 206 Problem. Let f : X Y be onetoone, onto mp between metric spces X, Y. () If f is continuous nd X is compct, prove tht f is homeomorphism. Does this result remin true
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 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 informationWeek 7 Riemann Stieltjes Integration: Lectures 1921
Week 7 Riemnn Stieltjes Integrtion: Lectures 1921 Lecture 19 Throughout this section α will denote monotoniclly incresing function on n intervl [, b]. Let f be bounded function on [, b]. Let P = { = 0
More informationMath 1431 Section M TH 4:00 PM 6:00 PM Susan Wheeler Office Hours: Wed 6:00 7:00 PM Online ***NOTE LABS ARE MON AND WED
Mth 43 Section 4839 M TH 4: PM 6: PM Susn Wheeler swheeler@mth.uh.edu Office Hours: Wed 6: 7: PM Online ***NOTE LABS ARE MON AND WED t :3 PM to 3: pm ONLINE Approimting the re under curve given the type
More information7.2 The Definition of the Riemann Integral. Outline
7.2 The Definition of the Riemnn Integrl Tom Lewis Fll Semester 2014 Upper nd lower sums Some importnt theorems Upper nd lower integrls The integrl Two importnt theorems on integrbility Outline Upper nd
More informationMath 360: A primitive integral and elementary functions
Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:
More informationThe Fundamental Theorem of Calculus
The Fundmentl Theorem of Clculus MATH 151 Clculus for Mngement J. Robert Buchnn Deprtment of Mthemtics Fll 2018 Objectives Define nd evlute definite integrls using the concept of re. Evlute definite integrls
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 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 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 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 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 informationMore Properties of the Riemann Integral
More Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologil Sienes nd Deprtment of Mthemtil Sienes Clemson University Februry 15, 2018 Outline More Riemnn Integrl Properties The Fundmentl
More informationRegulated functions and the regulated integral
Regulted functions nd the regulted integrl Jordn Bell jordn.bell@gmil.com Deprtment of Mthemtics University of Toronto April 3 2014 1 Regulted functions nd step functions Let = [ b] nd let X be normed
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 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 informationMATH , Calculus 2, Fall 2018
MATH 362, 363 Clculus 2, Fll 28 The FUNdmentl Theorem of Clculus Sections 5.4 nd 5.5 This worksheet focuses on the most importnt theorem in clculus. In fct, the Fundmentl Theorem of Clculus (FTC is rgubly
More information1 The fundamental theorems of calculus.
The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl new nme for ntiderivtive. Differentiting integrls. Tody we provide the connection
More informationEuler, Ioachimescu and the trapezium rule. G.J.O. Jameson (Math. Gazette 96 (2012), )
Euler, Iochimescu nd the trpezium rule G.J.O. Jmeson (Mth. Gzette 96 (0), 36 4) The following results were estblished in recent Gzette rticle [, Theorems, 3, 4]. Given > 0 nd 0 < s
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 informationIntegration Techniques
Integrtion Techniques. Integrtion of Trigonometric Functions Exmple. Evlute cos x. Recll tht cos x = cos x. Hence, cos x Exmple. Evlute = ( + cos x) = (x + sin x) + C = x + 4 sin x + C. cos 3 x. Let u
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 informationMath 324 Course Notes: Brief description
Brief description These re notes for Mth 324, n introductory course in Mesure nd Integrtion. Students re dvised to go through ll sections in detil nd ttempt ll problems. These notes will be modified nd
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 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 informationDefinite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +
Definite Integrls 5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the lefthnd
More informationSections 5.2: The Definite Integral
Sections 5.2: The Definite Integrl In this section we shll formlize the ides from the lst section to functions in generl. We strt with forml definition.. The Definite Integrl Definition.. Suppose f(x)
More informationProperties of the Riemann Stieltjes Integral
Properties of the Riemnn Stieltjes Integrl Theorem (Linerity Properties) Let < c < d < b nd A,B IR nd f,g,α,β : [,b] IR. () If f,g R(α) on [,b], then Af +Bg R(α) on [,b] nd [ ] b Af +Bg dα A +B (b) If
More informationChapter 1. Basic Concepts
Socrtes Dilecticl Process: The Þrst step is the seprtion of subject into its elements. After this, by deþning nd discovering more bout its prts, one better comprehends the entire subject Socrtes (469399)
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 information63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =
More informationFunctions of bounded variation
Division for Mthemtics Mrtin Lind Functions of bounded vrition Mthemtics Clevel thesis Dte: 20060130 Supervisor: Viktor Kold Exminer: Thoms Mrtinsson Krlstds universitet 651 88 Krlstd Tfn 054700 10
More information0.1 Properties of regulated functions and their Integrals.
MA244 Anlysis III Solutions. Sheet 2. NB. THESE ARE SKELETON SOLUTIONS, USE WISELY!. Properties of regulted functions nd their Integrls.. (Q.) Pick ny ɛ >. As f, g re regulted, there exist φ, ψ S[, b]:
More informationPhil Wertheimer UMD Math Qualifying Exam Solutions Analysis  January, 2015
Problem 1 Let m denote the Lebesgue mesure restricted to the compct intervl [, b]. () Prove tht function f defined on the compct intervl [, b] is Lipschitz if nd only if there is constct c nd function
More informationIndefinite Integral. Chapter Integration  reverse of differentiation
Chpter Indefinite Integrl Most of the mthemticl opertions hve inverse opertions. The inverse opertion of differentition is clled integrtion. For exmple, describing process t the given moment knowing the
More informationBest Approximation in the 2norm
Jim Lmbers MAT 77 Fll Semester 111 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the norm Suppose tht we wish to obtin function f n (x) tht is liner combintion
More informationQuestion 1. Question 3. Question 4. Graduate Analysis I Chapter 5
Grdute Anlysis I Chpter 5 Question If f is simple mesurle function (not necessrily positive) tking vlues j on j, j =,,..., N, show tht f = N j= j j. Proof. We ssume j disjoint nd,, J e nonnegtive ut J+,,
More informationNew Integral Inequalities for ntime Differentiable Functions with Applications for pdfs
Applied Mthemticl Sciences, Vol. 2, 2008, no. 8, 353362 New Integrl Inequlities for ntime Differentible Functions with Applictions for pdfs Aristides I. Kechriniotis Technologicl Eductionl Institute
More informationMAA 4212 Improper Integrals
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
More informationEntrance Exam, Real Analysis September 1, 2009 Solve exactly 6 out of the 8 problems. Compute the following and justify your computation: lim
1. Let n be positive integers. ntrnce xm, Rel Anlysis September 1, 29 Solve exctly 6 out of the 8 problems. Sketch the grph of the function f(x): f(x) = lim e x2n. Compute the following nd justify your
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 informationFundamental Theorem of Calculus
Fundmentl Theorem of Clculus Recll tht if f is nonnegtive nd continuous on [, ], then the re under its grph etween nd is the definite integrl A= f() d Now, for in the intervl [, ], let A() e the re under
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 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 informationAnalysis Comp Study Guide
Anlysis Comp Study Guide The Rel nd Complex Number Systems nd Bsic Topology Theorem 1 (CuchySchwrz Inequlity). ( n ) 2 k b k b 2 k. 2 k As ( k tb k ) 2 0, s qudrtic in t it hs t most one root. So the
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 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 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 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 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 informationCzechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction
Czechoslovk Mthemticl Journl, 55 (130) (2005), 933 940 ESTIMATES OF THE REMAINDER IN TAYLOR S THEOREM USING THE HENSTOCKKURZWEIL INTEGRAL, Abbotsford (Received Jnury 22, 2003) Abstrct. When relvlued
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 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 informationarxiv: v1 [math.ca] 7 Mar 2012
rxiv:1203.1462v1 [mth.ca] 7 Mr 2012 A simple proof of the Fundmentl Theorem of Clculus for the Lebesgue integrl Mrch, 2012 Rodrigo López Pouso Deprtmento de Análise Mtemátic Fcultde de Mtemátics, Universidde
More information