Properties of the Riemann Stieltjes Integral


 Tracy Cross
 8 months ago
 Views:
Transcription
1 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 f R(α) R(β) on [,b], then f R(Aα+Bβ) on [,b] nd f d(aα+bβ) A +B (c) If f R(α) on [,c] nd on [c,b], then f R(α) on [,b] nd c + (d) If f R(α) on [,b] then f R(α) on [c,d] [,b]. c f dβ Proof: () Let ε > 0. Then, for ny prtition IP of [,b] nd choice T for IP, S(IP,T,Af +Bg,α) A B AS(IP,T,f,α)+BS(IP,T,g,α) A B A S(IP,T,f,α) + B S(IP,T,g,α) Assume tht A nd B re nonzero. (The cses tht A nd/or B re zero re similr, but esier.) Since f R(α) on [,b] there is prtition IP f,ε such tht S(IP,T,f,α) < ε whenever IP IP f,ε 2 A nd since g R(α) on [,b] there is prtition IP g,ε such tht S(IP,T,g,α) < ε whenever IP IP g,ε 2 B It now suffices to set IP ε IP f,ε IP g,ε nd observe tht S(IP,T,Af +Bg,α) A B < ε (b) See Problem Set 1, #3. (c) See Problem Set 1, #2. (d) See Problem Set 2, #3. whenever IP IP ε c Joel Feldmn All rights reserved. Jnury 20, 2017 Properties of the Riemnn Stieltjes Integrl 1
2 Theorem (Integrtion by Prts) Let < b nd f,α : [,b] IR. If f R(α) on [,b], then α R(f) on [,b] nd f(x)dα(x)+ α(x)df(x) f(b)α(b) f()α() Remrk. () The integrtion by prts formul my lso be written + αdf d(fα) (b) We shll lter see tht if α hs continuous first derivtive, then f(x)dα(x) f(x)α (x)dx. So if both f nd α hve continuous first derivtives, we my write the integrtion by prts formul s f(x)α (x)dx f(b)α(b) f()α() α(x)f (x)dx which is the integrtion by prts formul of first yer clculus (though you probbly used f(x) u(x) nd α(x) v(x). Proof of integrtion by prts: Our gol is to show tht αdf exists nd tkes the vlue αdf f(b)α(b) f()α() So let s look t the difference between Riemnn sum for αdf nd the right hnd side. For ny prtition IP { x 0, x 1, x 2, x 3,, x n b of [,b] nd choice T { t 1, t 2, t 3,, t n for IP, Note tht S ( IP,T,α,f ) { f(b)α(b) f()α() α(t i ) [ ] n f(x i ) f(x i 1 α(x i )f(x i ) + {{ α(b)f(b) for in α(x i 1 )f(x i 1 ) + {{ α()f() for (The 1 i n 1 terms of the second sum cncel the 2 i n terms of the third sum.) f(x i ) [ α(x i ) α(t i ) ] f(x j 1 ) [ α(t j ) α(x j 1 ) ] + j1 c Joel Feldmn All rights reserved. Jnury 20, 2017 Properties of the Riemnn Stieltjes Integrl 2
3 f(x i ) [ α(x i ) α(t i ) ] looks like term in Riemnn sum pproximtion to with subintervl [t i,x i ] nd choice point x i [t i,x i ] nd f(x j 1 ) [ α(t j ) α(x j 1 ) ] looks like term in Riemnn sum pproximtion to with subintervl [x j 1,t j ] nd choice point x j 1 [x j 1,t j ]. Here is figure which shows ll of these subintervls. x 0 j 1 i 1 j 2 i i n 1 j i n t 1 x 1 t 2 2x 2 t n 1 x n 1 nt n x n b So ll of the subintervls fit together perfectly to form the prtition 1 nd where the choice IP T { x 0, t 1, x 1, t 2, x 2, t 3,, x n 1, t n, x n S ( IP,T,α,f ) T { j1 x 0, x 1, { f(b)α(b) f()α() j2 x 1, i2 x 2, S ( IP T,T,f,α ) + j3 x 2, i3 in 1 x 3,, x n 1, jn x n 1, Now let ε > 0. Since f R(α) for [,b] there is prtition IP ε such tht S( ) b IP, T,f,α < ε in x n for ll prtitions IP finer thn IP ε. If the prtitition IP bove is finer thn IP ε then the prtition IP T is lso finer thn IP ε nd we hve S( IP,T,α,f ) { f(b)α(b) f()α() S( IP T,T,f,α ) < ε Theorem (The Chnge of Vribles x g(y)) Let < b nd c < d, g : [c,d] [,b] be continuous, strictly monotoniclly incresing, nd obey g(c) nd g(d) b nd f,α : [,b] IR. Set h(y) f ( g(y) ) β(y) α ( g(y) ) If f R(α) on [,b], then h R(β) on [c,d] nd d c h(y)dβ(y) f(x)dα(x) 1 There is subtlety hidden in these definitions. We re not llowed to use subintervls of width zero. So if, for exmple, t i x i, we merge the two subintervls [x i 1,t i ], [t i,x i ] into the single subintervl [x i 1,x i ] nd use f(x i 1 )[α(t i ) α(x i 1 )]+f(x i )[α(x i ) α(t i )] f(x i 1 )[α(x i ) α(x i 1 )]. c Joel Feldmn All rights reserved. Jnury 20, 2017 Properties of the Riemnn Stieltjes Integrl 3
4 Proof: Our gol is to prove tht d c h(y)dβ(y) exists nd equls f(x)dα(x), so let s consider the difference between Riemnn sum for d c h(y)dβ(y) nd f(x)dα(x). For ny prtition IP {c y 0,,y n d of [c,d] nd ny choice T {s 1,,s n for IP S(IP,T,h,β) n h(s i ) [ β(y i ) β(y i 1 ) ] f ( g(s i ) )[ α ( g(y i ) ) α ( g(y i 1 ) )] S(g(IP),g(T),f,α) where g(ip) { g(y) y IP { x0 g(y 0 ) g(c), x 1 g(y 1 ),, x n g(y n ) g(d) b is prtition of [,b] becuse g is ssumed to be strictly monotonic, so tht y i 1 < y i x i 1 < x i nd is ssumed to obey x 0 g(y 0 ) nd x n g(y n ) b nd g(t) { t1 g(s 1 ),, t n g(s n ) is choice for g(ip) becuse g is ssumed to be strictly monotonic so tht y i 1 s i y i x i 1 g(y i 1 ) g(s i ) t i g(y i ) x i Now let ε > 0. We hve ssumed tht f R(α) on [,b]. So there is prtition IP f,ε of [,b] such tht IP f IP f,ε S(IP f,t f,f,α) < ε for ll choices T f for IP f. Thessumptionsthtwehvemdeong gurnteethttheinversefunctiong 1 : [,b] [c,d] exists nd tht g 1( IP f,ε ) is prtition of [c,d]. We choose IPε g 1( IP f,ε ). Then s desired. IP IP ε g(ip) g(ip ε ) IP f,ε S(IP,T,h,β) S( finer thn IP f,ε g(ip), llowed T f g(t),f,α) < ε c Joel Feldmn All rights reserved. Jnury 20, 2017 Properties of the Riemnn Stieltjes Integrl 4
5 Theorem (Second Fundmentl Theorem of Clculus) Let < b nd f : [, b] IR. Assume tht f is differentible on [,b] nd f R on [,b] Then f (x)dx f(b) f() Proof: Let IP { x 0,x 1,x 2,,x n be ny prtition of [,b]. Then, by the men vlue theorem, there exists, for ech 1 i n t i [x i 1,x i ] with So, setting T { t 1,t 2,,t n, we hve S(IP,T,f ) f(x i ) f(x i 1 ) f (t i ) ( x i x i 1 ) f (t i ) ( ) n [ x i x i 1 f(xi ) f(x i 1 ) ] f(b) f() So now we just hve to pply the definition of f R on [,b]. Theorem (Bsic Bounds) Let < b nd f,g,α : [,b] IR. Assume tht f,g R(α) on [,b] nd α is monotoniclly incresing. () If f(x) g(x) for ll x [,b], then () If f(x) g(x) for ll x [,b], then Proof: Let ε > 0. Since f R(α) on [,b] there is prtition IP f,ε such tht S(IP,T,f,α) < ε whenever IP IP f,ε nd T is choice for IP. Since g R(α) on [,b] there is prtition IP g,ε such tht b S(IP,T,g,α) < ε whenever IP IP g,ε nd T is choice for IP. Set IP ε { x 0,x 1,x 2,,x n b IP f,ε IP g,ε nd let T { t 1,t 2,,t n be choice for IP ε. c Joel Feldmn All rights reserved. Jnury 20, 2017 Properties of the Riemnn Stieltjes Integrl 5
6 () We hve S(IP ε,t,f,α)+ε f(t i ) [ α(x i ) α(x i 1 ) ] +ε g(t i ) [ α(x i ) α(x i 1 ) ] +ε (since f(t i ) g(t i ) nd α(x i ) α(x i 1 ) 0) S(IP ε,t,g,α)+ε +2ε As +2ε is true for ll ε > 0, we lso hve. (b) We hve As S(IPε,T,f,α) +ε f(ti ) [ α(x i ) α(x i 1 ) ] +ε g(t i ) [ α(x i ) α(x i 1 ) ] +ε (since f(t i ) g(t i ) nd α(x i ) α(x i 1 ) 0) S(IP ε,t,g,α)+ε +2ε +2ε is true for ll ε > 0, we lso hve. Theorem (First Fundmentl Theorem of Clculus) Let < b nd f : [,b] IR. Assume tht f R on [,b] Set, for x b, Then () F is continuous on [, b] nd F(x) x f(x)dx (b) if f is continuous t x 0 [,b], then F is differentible t x 0 nd F (x 0 ) f(x 0 ). c Joel Feldmn All rights reserved. Jnury 20, 2017 Properties of the Riemnn Stieltjes Integrl 6
7 Proof: () Since f R it is bounded. Suppose tht f(t) M for ll t b. Then F(y) F(x) y x f(t)dt M y x so F is uniformly continous. (b) Let f be continuous t x 0 [,b]. Then, for ll x b F(x) F(x 0 ) x f(x 0 ) x x 0 x 0 f(t)dt f(x 0 )[x x 0 ] x x 0 1 x [ f(t) f(x0 ) ] dt x x 0 x 0 sup f(t) f(x0 ) t between x 0 nd x Since f is continuous t x 0, the right hnd side converges to zero s x x 0. c Joel Feldmn All rights reserved. Jnury 20, 2017 Properties of the Riemnn Stieltjes Integrl 7
Math 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 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 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 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 information7  Continuous random variables
71 Continuous rndom vribles S. Lll, Stnford 2011.01.25.01 7  Continuous rndom vribles Continuous rndom vribles The cumultive distribution function The uniform rndom vrible Gussin rndom vribles The Gussin
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 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 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 informationThe RiemannStieltjes Integral
Chpter 6 The RiemnnStieltjes Integrl 6.1. Definition nd Eistene of the Integrl Definition 6.1. Let, b R nd < b. ( A prtition P of intervl [, b] is finite set of points P = { 0, 1,..., n } suh tht = 0
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 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 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 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 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 informationLine Integrals. Partitioning the Curve. Estimating the Mass
Line Integrls Suppose we hve curve in the xy plne nd ssocite density δ(p ) = δ(x, y) t ech point on the curve. urves, of course, do not hve density or mss, but it my sometimes be convenient or useful to
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 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 informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationLecture 14: Quadrature
Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be relvlues nd smooth The pproximtion of n integrl by numericl
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 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 informationNUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.
NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with
More informationPolynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230
Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given
More informationa n+2 a n+1 M n a 2 a 1. (2)
Rel Anlysis Fll 004 Tke Home Finl Key 1. Suppose tht f is uniformly continuous on set S R nd {x n } is Cuchy sequence in S. Prove tht {f(x n )} is Cuchy sequence. (f is not ssumed to be continuous outside
More informationSection 4.8. D v(t j 1 ) t. (4.8.1) j=1
Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions
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 informationSection 6.1 Definite Integral
Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined
More informationMA Handout 2: Notation and Background Concepts from Analysis
MA350059 Hndout 2: Nottion nd Bckground Concepts from Anlysis This hndout summrises some nottion we will use nd lso gives recp of some concepts from other units (MA20023: PDEs nd CM, MA20218: Anlysis 2A,
More informationPractice final exam solutions
University of Pennsylvni Deprtment of Mthemtics Mth 26 Honors Clculus II Spring Semester 29 Prof. Grssi, T.A. Asher Auel Prctice finl exm solutions 1. Let F : 2 2 be defined by F (x, y (x + y, x y. If
More informationNecessary and Sufficient Conditions for Differentiating Under the Integral Sign
Necessry nd Sufficient Conditions for Differentiting Under the Integrl Sign Erik Tlvil 1. INTRODUCTION. When we hve n integrl tht depends on prmeter, sy F(x f (x, y dy, it is often importnt to know when
More informationKOÇ UNIVERSITY MATH 106 FINAL EXAM JANUARY 6, 2013
KOÇ UNIVERSITY MATH 6 FINAL EXAM JANUARY 6, 23 Durtion of Exm: 2 minutes INSTRUCTIONS: No clcultors nd no cell phones my be used on the test. No questions, nd tlking llowed. You must lwys explin your nswers
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 informationThe problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests.
ADVANCED CALCULUS PRACTICE PROBLEMS JAMES KEESLING The problems tht follow illustrte the methods covered in clss. They re typicl of the types of problems tht will be on the tests. 1. Riemnn Integrtion
More informationAnonymous Math 361: Homework 5. x i = 1 (1 u i )
Anonymous Mth 36: Homewor 5 Rudin. Let I be the set of ll u (u,..., u ) R with u i for ll i; let Q be the set of ll x (x,..., x ) R with x i, x i. (I is the unit cube; Q is the stndrd simplex in R ). Define
More informationThe Wave Equation I. MA 436 Kurt Bryan
1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string
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 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 information(0.0)(0.1)+(0.3)(0.1)+(0.6)(0.1)+ +(2.7)(0.1) = 1.35
7 Integrtion º½ ÌÛÓ Ü ÑÔÐ Up to now we hve been concerned with extrcting informtion bout how function chnges from the function itself. Given knowledge bout n object s position, for exmple, we wnt to know
More informationCalculus MATH 172Fall 2017 Lecture Notes
Clculus MATH 172Fll 2017 Lecture Notes These notes re concise summry of wht hs been covered so fr during the lectures. All the definitions must be memorized nd understood. Sttements of importnt theorems
More informationLecture 14 Numerical integration: advanced topics
Lecture 14 Numericl integrtion: dvnced topics Weinn E 1,2 nd Tiejun Li 2 1 Deprtment of Mthemtics, Princeton University, weinn@princeton.edu 2 School of Mthemticl Sciences, Peking University, tieli@pku.edu.cn
More informationCalculus 2: Integration. Differentiation. Integration
Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is
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 informationFINALTERM EXAMINATION 2011 Calculus &. Analytical GeometryI
FINALTERM EXAMINATION 011 Clculus &. Anlyticl GeometryI Question No: 1 { Mrks: 1 )  Plese choose one If f is twice differentible function t sttionry point x 0 x 0 nd f ''(x 0 ) > 0 then f hs reltive...
More informationUnit #10 De+inite Integration & The Fundamental Theorem Of Calculus
Unit # De+inite Integrtion & The Fundmentl Theorem Of Clculus. Find the re of the shded region ove nd explin the mening of your nswer. (squres re y units) ) The grph to the right is f(x) = x + 8x )Use
More information2 Definitions and Basic Properties of Extended Riemann Stieltjes Integrals
2 Definitions nd Bsic Properties of Extended Riemnn Stieltjes Integrls 2.1 Regulted nd Intervl Functions Regulted functions Let X be Bnch spce, nd let J be nonempty intervl in R, which my be bounded or
More informationapproaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below
. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.
More information3.4 Numerical integration
3.4. Numericl integrtion 63 3.4 Numericl integrtion In mny economic pplictions it is necessry to compute the definite integrl of relvlued function f with respect to "weight" function w over n intervl [,
More informationSection 14.3 Arc Length and Curvature
Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in
More information7.6 The Use of Definite Integrals in Physics and Engineering
Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 7.6 The Use of Definite Integrls in Physics nd Engineering It hs been shown how clculus cn be pplied to find solutions to geometric problems
More informationHenstock Kurzweil delta and nabla integrals
Henstock Kurzweil delt nd nbl integrls Alln Peterson nd Bevn Thompson Deprtment of Mthemtics nd Sttistics, University of NebrskLincoln Lincoln, NE 685880323 peterso@mth.unl.edu Mthemtics, SPS, The University
More informationSTEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.
STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t
More informationu(t)dt + i a f(t)dt f(t) dt b f(t) dt (2) With this preliminary step in place, we are ready to define integration on a general curve in C.
Lecture 4 Complex Integrtion MATHGA 2451.001 Complex Vriles 1 Construction 1.1 Integrting complex function over curve in C A nturl wy to construct the integrl of complex function over curve in the complex
More informationLecture 17. Integration: Gauss Quadrature. David Semeraro. University of Illinois at UrbanaChampaign. March 20, 2014
Lecture 17 Integrtion: Guss Qudrture Dvid Semerro University of Illinois t UrbnChmpign Mrch 0, 014 Dvid Semerro (NCSA) CS 57 Mrch 0, 014 1 / 9 Tody: Objectives identify the most widely used qudrture method
More informationS. S. Dragomir. 1. Introduction. In [1], Guessab and Schmeisser have proved among others, the following companion of Ostrowski s inequality:
FACTA UNIVERSITATIS NIŠ) Ser Mth Inform 9 00) 6 SOME COMPANIONS OF OSTROWSKI S INEQUALITY FOR ABSOLUTELY CONTINUOUS FUNCTIONS AND APPLICATIONS S S Drgomir Dedicted to Prof G Mstroinni for his 65th birthdy
More informationarxiv: v1 [math.ca] 9 Jun 2011
Men vlue integrl inequlities rxiv:1106.1807v1 [mth.ca] 9 Jun 2011 June, 2011 Rodrigo López Pouso Deprtment of Mthemticl Anlysis Fculty of Mthemtics, University of Sntigo de Compostel, 15782 Sntigo de Compostel,
More information1 Error Analysis of Simple Rules for Numerical Integration
cs41: introduction to numericl nlysis 11/16/10 Lecture 19: Numericl Integrtion II Instructor: Professor Amos Ron Scries: Mrk Cowlishw, Nthnel Fillmore 1 Error Anlysis of Simple Rules for Numericl Integrtion
More informationIf u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then f(g(x))g (x) dx = f(u) du
Integrtion by Substitution: The Fundmentl Theorem of Clculus demonstrted the importnce of being ble to find ntiderivtives. We now introduce some methods for finding ntiderivtives: If u = g(x) is differentible
More informationC1M14. Integrals as Area Accumulators
CM Integrls s Are Accumultors Most tetbooks do good job of developing the integrl nd this is not the plce to provide tht development. We will show how Mple presents Riemnn Sums nd the ccompnying digrms
More informationCS667 Lecture 6: Monte Carlo Integration 02/10/05
CS667 Lecture 6: Monte Crlo Integrtion 02/10/05 Venkt Krishnrj Lecturer: Steve Mrschner 1 Ide The min ide of Monte Crlo Integrtion is tht we cn estimte the vlue of n integrl by looking t lrge number of
More informationMath 0230 Calculus 2 Lectures
Mth Clculus Lectures Chpter 7 Applictions of Integrtion Numertion of sections corresponds to the text Jmes Stewrt, Essentil Clculus, Erly Trnscendentls, Second edition. Section 7. Ares Between Curves Two
More information38.2. The Uniform Distribution. Introduction. Prerequisites. Learning Outcomes
The Uniform Distribution 8. Introduction This Section introduces the simplest type of continuous probbility distribution which fetures continuous rndom vrible X with probbility density function f(x) which
More informationWe know that if f is a continuous nonnegative function on the interval [a, b], then b
1 Ares Between Curves c 22 Donld Kreider nd Dwight Lhr We know tht if f is continuous nonnegtive function on the intervl [, b], then f(x) dx is the re under the grph of f nd bove the intervl. We re going
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 informationNumerical integration
2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter
More informationLecture 1  Introduction and Basic Facts about PDEs
* 18.15  Introdution to PDEs, Fll 004 Prof. Gigliol Stffilni Leture 1  Introdution nd Bsi Fts bout PDEs The Content of the Course Definition of Prtil Differentil Eqution (PDE) Liner PDEs VVVVVVVVVVVVVVVVVVVV
More informationA Convergence Theorem for the Improper Riemann Integral of Banach Spacevalued Functions
Interntionl Journl of Mthemticl Anlysis Vol. 8, 2014, no. 50, 24512460 HIKARI Ltd, www.mhikri.com http://dx.doi.org/10.12988/ijm.2014.49294 A Convergence Theorem for the Improper Riemnn Integrl of Bnch
More informationTHE HANKEL MATRIX METHOD FOR GAUSSIAN QUADRATURE IN 1 AND 2 DIMENSIONS
THE HANKEL MATRIX METHOD FOR GAUSSIAN QUADRATURE IN 1 AND 2 DIMENSIONS CARLOS SUERO, MAURICIO ALMANZAR CONTENTS 1 Introduction 1 2 Proof of Gussin Qudrture 6 3 Iterted 2Dimensionl Gussin Qudrture 20 4
More informationBest Approximation. Chapter The General Case
Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given
More information1 Sequences. 2 Series. 2 SERIES Analysis Study Guide
2 SERIES Anlysis Study Guide 1 Sequences Def: An ordered field is field F nd totl order < (for ll x, y, z F ): (i) x < y, y < x or x = y, (ii) x < y, y < z x < z (iii) x < y x + z < y + z (iv) 0 < y, x
More information(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P.
Chpter 7: The Riemnn Integrl When the derivtive is introdued, it is not hrd to see tht the it of the differene quotient should be equl to the slope of the tngent line, or when the horizontl xis is time
More informationON THE INEQUALITY OF THE DIFFERENCE OF TWO INTEGRAL MEANS AND APPLICATIONS FOR PDFs
ON THE INEQUALITY OF THE DIFFERENCE OF TWO INTEGRAL MEANS AND APPLICATIONS FOR PDFs A.I. KECHRINIOTIS AND N.D. ASSIMAKIS Deprtment of Eletronis Tehnologil Edutionl Institute of Lmi, Greee EMil: {kehrin,
More informationLecture 6: Singular Integrals, Open Quadrature rules, and Gauss Quadrature
Lecture notes on Vritionl nd Approximte Methods in Applied Mthemtics  A Peirce UBC Lecture 6: Singulr Integrls, Open Qudrture rules, nd Guss Qudrture (Compiled 6 August 7) In this lecture we discuss the
More informationON SOME NEW FRACTIONAL INTEGRAL INEQUALITIES
Volume 1 29, Issue 3, Article 86, 5 pp. ON SOME NEW FRACTIONAL INTEGRAL INEQUALITIES SOUMIA BELARBI AND ZOUBIR DAHMANI DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MOSTAGANEM soumimth@hotmil.fr zzdhmni@yhoo.fr
More informationMath 4200: Homework Problems
Mth 4200: Homework Problems Gregor Kovčič 1. Prove the following properties of the binomil coefficients ( n ) ( n ) (i) 1 + + + + 1 2 ( n ) (ii) 1 ( n ) ( n ) + 2 + 3 + + n 2 3 ( ) n ( n + = 2 n 1 n) n,
More informationMATH 174A: PROBLEM SET 5. Suggested Solution
MATH 174A: PROBLEM SET 5 Suggested Solution Problem 1. Suppose tht I [, b] is n intervl. Let f 1 b f() d for f C(I; R) (i.e. f is continuous relvlued function on I), nd let L 1 (I) denote the completion
More informationA basic logarithmic inequality, and the logarithmic mean
Notes on Number Theory nd Discrete Mthemtics ISSN 30 532 Vol. 2, 205, No., 3 35 A bsic logrithmic inequlity, nd the logrithmic men József Sándor Deprtment of Mthemtics, BbeşBolyi University Str. Koglnicenu
More informationUniversitaireWiskundeCompetitie. Problem 2005/4A We have k=1. Show that for every q Q satisfying 0 < q < 1, there exists a finite subset K N so that
Problemen/UWC NAW 5/7 nr juni 006 47 Problemen/UWC UniversitireWiskundeCompetitie Edition 005/4 For Session 005/4 we received submissions from Peter Vndendriessche, Vldislv Frnk, Arne Smeets, Jn vn de
More informationIII. Lecture on Numerical Integration. File faclib/dattab/lecturenotes/numericalinter03.tex /by EC, 3/14/2008 at 15:11, version 9
III Lecture on Numericl Integrtion File fclib/dttb/lecturenotes/numericalinter03.tex /by EC, 3/14/008 t 15:11, version 9 1 Sttement of the Numericl Integrtion Problem In this lecture we consider the
More informationLine Integrals and Entire Functions
Line Integrls nd Entire Funtions Defining n Integrl for omplex Vlued Funtions In the following setions, our min gol is to show tht every entire funtion n be represented s n everywhere onvergent power series
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 informationMTH 122 Fall 2008 Essex County College Division of Mathematics Handout Version 10 1 October 14, 2008
MTH 22 Fll 28 Essex County College Division of Mthemtics Hndout Version October 4, 28 Arc Length Everyone should be fmilir with the distnce formul tht ws introduced in elementry lgebr. It is bsic formul
More informationOn the Generalized Weighted QuasiArithmetic Integral Mean 1
Int. Journl of Mth. Anlysis, Vol. 7, 2013, no. 41, 20392048 HIKARI Ltd, www.mhikri.com http://dx.doi.org/10.12988/ijm.2013.3499 On the Generlized Weighted QusiArithmetic Integrl Men 1 Hui Sun School
More information5: The Definite Integral
5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce
More informationKeywords : Generalized Ostrowski s inequality, generalized midpoint inequality, Taylor s formula.
Generliztions of the Ostrowski s inequlity K. S. Anstsiou Aristides I. Kechriniotis B. A. Kotsos Technologicl Eductionl Institute T.E.I.) of Lmi 3rd Km. O.N.R. LmiAthens Lmi 3500 Greece Abstrct Using
More informationMath RE  Calculus II Area Page 1 of 12
Mth RE  Clculus II re Pge of re nd the Riemnn Sum Let f) be continuous function nd = f) f) > on closed intervl,b] s shown on the grph. The Riemnn Sum theor shows tht the re of R the region R hs re=
More informationl 2 p2 n 4n 2, the total surface area of the
Week 6 Lectures Sections 7.5, 7.6 Section 7.5: Surfce re of Revolution Surfce re of Cone: Let C be circle of rdius r. Let P n be n nsided regulr polygon of perimeter p n with vertices on C. Form cone
More information13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS
33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in
More information31.2. Numerical Integration. Introduction. Prerequisites. Learning Outcomes
Numericl Integrtion 3. Introduction In this Section we will present some methods tht cn be used to pproximte integrls. Attention will be pid to how we ensure tht such pproximtions cn be gurnteed to be
More informationNOTES AND PROBLEMS: INTEGRATION THEORY
NOTES AND PROBLEMS: INTEGRATION THEORY SAMEER CHAVAN Abstrct. These re the lecture notes prepred for prticipnts of AFSI to be conducted t Kumun University, Almor from 1st to 27th December, 2014. Contents
More informationTopic 1 Notes Jeremy Orloff
Topic 1 Notes Jerem Orloff 1 Introduction to differentil equtions 1.1 Gols 1. Know the definition of differentil eqution. 2. Know our first nd second most importnt equtions nd their solutions. 3. Be ble
More informationNotes on Calculus II Integral Calculus. Miguel A. Lerma
Notes on Clculus II Integrl Clculus Miguel A. Lerm November 22, 22 Contents Introduction 5 Chpter. Integrls 6.. Ares nd Distnces. The Definite Integrl 6.2. The Evlution Theorem.3. The Fundmentl Theorem
More information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationTest , 8.2, 8.4 (density only), 8.5 (work only), 9.1, 9.2 and 9.3 related test 1 material and material from prior classes
Test 2 8., 8.2, 8.4 (density only), 8.5 (work only), 9., 9.2 nd 9.3 relted test mteril nd mteril from prior clsses Locl to Globl Perspectives Anlyze smll pieces to understnd the big picture. Exmples: numericl
More informationChapter 1  Functions and Variables
Business Clculus 1 Chpter 1  Functions nd Vribles This Acdemic Review is brought to you free of chrge by preptests4u.com. Any sle or trde of this review is strictly prohibited. Business Clculus 1 Ch 1:
More informationPDE Notes. Paul Carnig. January ODE s vs PDE s 1
PDE Notes Pul Crnig Jnury 2014 Contents 1 ODE s vs PDE s 1 2 Section 1.2 Het diffusion Eqution 1 2.1 Fourier s w of Het Conduction............................. 2 2.2 Energy Conservtion.....................................
More informationThe Definite Integral
CHAPTER 3 The Definite Integrl Key Words nd Concepts: Definite Integrl Questions to Consider: How do we use slicing to turn problem sttement into definite integrl? How re definite nd indefinite integrls
More informationMATH 1080: Calculus of One Variable II Fall 2017 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart.
MATH 1080: Clculus of One Vrile II Fll 2017 Textook: Single Vrile Clculus: Erly Trnscendentls, 7e, y Jmes Stewrt Unit 2 Skill Set Importnt: Students should expect test questions tht require synthesis of
More informationMath 113 Exam 2 Practice
Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number
More information