DEFINITE INTEGRALS. f(x)dx exists. Note that, together with the definition of definite integrals, definitions (2) and (3) define b


 Bruno Young
 2 years ago
 Views:
Transcription
1 DEFINITE INTEGRALS JOHN D. MCCARTHY Astrct. These re lecture notes for Sections 5.3 nd Section 5.3 Definition 1. f is integrle on [, ] if f(x)dx exists. Definition 2. If f() is defined, then f(x)dx. = 0. Definition 3. If f is integrle on [, ], then Note tht, together with the definition of definite integrls, definitions (2) nd (3) define f(x)dx whenever nd re contined in some intervl I contined in the domin of f or when = nd is in the domin of f. Theorem 1. The Existence of Definite Integrls If f is continuous on [, ], then f is integrle on [, ]. Proof. The proof of this theorem uses methods not covered in MTH 132. Theorem 2. Order of Integrtion If f is integrle on n intervl I nd, I, then Proof. First, consider the cse where <. Then, y definition, Secondly, consider the cse where =. Then, y definition (2), 0 nd 0 = 0 nd, hence, Thirdly, consider the cse where >. Then, y definition (3), f(x)dx nd, hence, ( f(x)dx) = Thus, in ll cses, Theorem 3. Zero Width Intervl If f() is defined, then f(x) = 0. Proof. By definition (2), 0. Theorem 4. Constnt Multiple If f is integrle on n intervl I nd, I, then k k Dte: Novemer 28,
2 2 J. MCCARTHY Proof. First, consider the cse where <. Let P : x 0 <... < x n e prtition of [, ] nd c i [x i 1, x i ], 1 i n. Let: (1) S. = nd: (2) R. = so tht: kf(c i )( (x)) i f(c i )( (x)) i (3) nd: k lim S (4) It follows from equtions (1) nd (2) tht: lim R. (5) S = kr. (6) Hence, y equtions (3), (4), nd (5): k lim S = lim kr = k lim R = k Secondly, consider the cse where =. Then, y definition (2), k k 0 nd k k k0 = 0. Hence, k k Thirdly, consider the cse where >. Then y the first cse nd definition (3), k k k k Corollry 5. Additive Inverse If f is integrle on n intervl I nd, I, then Proof. By Theorem 4: (7) ( 1) ( 1) Theorem 6. Sum If f nd g re integrle on n intervl I nd, I, then (f(x) + g(x))dx = g(x)dx.
3 DEFINITE INTEGRALS 3 Proof. First consider the cse where <. Let P : x 0 <... < x n e prtition of [, ] nd c i [x i 1, x i ], 1 i n. Let: (8) R. = (f(c i ) + g(c i ))( (x)) i, (9) S. = nd: (10) T. = so tht: f(c i )( (x)) i, g(c i )( (x)) i (11) (12) nd: (f(x) + g(x))dx = lim R, lim S, (13) g(x)dx = lim T. It follows from equtions (8), (9), nd (10) tht: (14) R = S + T. (15) Hence, y the sum rule for limits nd equtions (11), (12), (13), nd (14): (f(x) + g(x))dx = lim R = lim S + T = lim S + = g(x)dx. lim T Secondly, consider the cse where =. Then, y definition (2), (f(x) + g(x))dx = (f(x) + g(x))dx = 0 nd g(x)dx = g(x)dx = = 0. Hence, (f(x) + g(x))dx = g(x)dx. Thirdly, consider the cse where >. Then, y the first cse nd definition (3), (f(x) + g(x))dx = (f(x) + g(x))dx = ( g(x)dx) = g(x)dx. Corollry 7. Difference If f nd g re integrle on n intervl I nd, I, then (f(x) g(x))dx = f(x)dx g(x)dx.
4 4 J. MCCARTHY Proof. It follows from Theorems 4 nd 6 tht: (16) (f(x) g(x))dx = (f(x) + ( 1)g(x))dx = = ( 1)g(x)dx = f(x)dx ( 1) g(x)dx. g(x)dx Theorem 8. Additivity If f is integrle on n intervl I nd,, c I, then Proof. Note tht there re severl cses to consider sed upon whether (i) <, =, or > ; (ii) < c, = c, or > c; nd (iii) < c, = c, or > c. Cse 1: < < c. Let P : x 0 <... < x m e prtition of [, ] nd Q : y 0 <... < y n e prtition of [, c]; c i [x i 1, x i ], 1 i m; nd d j [y j 1, y j ], 1 j n. Then let: (17) S. = nd: (18) T. = so tht: m f(c i )( (x)) i, f(d j )( (y)) j, j=1 (19) nd: lim S (20) lim T. Q 0 Now suppose tht ɛ is positive rel numer. By equtions (19) nd (20), there exist positive rel numers α nd β such tht if P < α nd Q < β, then: (21) S nd: (22) T f(x)dx < ɛ f(x)dx < ɛ. Note tht P Q : x 0 <... < x m = = y 0 <... < y n is prtition of [, c]. Let:
5 DEFINITE INTEGRALS 5 (23) R. = S + T = m f(c i )( (x)) i + f(d j )( (y)) j. Let δ = min{α, β}. Since α nd β re positive rel numers, δ is positive rel numer. Now suppose tht P Q < δ. Note tht P Q = mx{ P, Q }. Hence, P P Q < δ α nd Q P Q < δ β. It follows from equtions (21), (22), nd (23) tht: (24) R (. This implies tht: S f(x)dx) = (S f(x)dx + T (25) lim P Q 0 R = j=1 f(x)dx) + (T f(x)dx < ɛ + ɛ = 2ɛ. f(x)dx) Note tht R is Riemnn sum pproximtion for f(x)dx corresponding to the prtition P Q of [, c]. Thus: (26) lim R. P Q 0 It follows from equtions (25) nd (26) tht We shll now deduce the remining cses from Cse 1 nd the definitions of definite integrls. Cse 2: =. In this cse, y definition (2), 0. Hence, since = : (27) 0 + Henceforth, we ssume tht. Cse 3 = c. In this cse, y definition (2), 0. Moreover, y Theorem 2, f(x)dx, so tht 0. It follows tht Henceforth, we ssume tht c. Cse 4: = c. In this cse, y definition (2), (28) 0. Hence, since = c: Henceforth, we ssume tht c. 0 =
6 6 J. MCCARTHY Cse 5 < c <. By ssumption, f(x)dx, f(x)dx, nd f(x)dx exist. Hence, y Theorem 2, f(x)dx, c f(x)dx, nd f(x)dx exist. It follows from Cse 1 tht: (29) Tht is to sy: (30) nd, hence: c f(x)dx f(x)dx (31) Cse 6 < < c. By ssumption, f(x)dx, f(x)dx, nd f(x)dx exist. Hence, y Theorem 2, f(x)dx, f(x)dx, nd f(x)dx exist. It follows from Cse 1 tht: (32) Tht is to sy: (33) nd, hence: f(x)dx (34) Cse 7 c < <. By ssumption, f(x)dx, f(x)dx, nd f(x)dx exist. Hence, y Theorem 2, c f(x)dx, f(x)dx, nd c f(x)dx exist. It follows from Cse 1 tht: (35) Tht is to sy: c c (36) nd, hence: f(x)dx
7 DEFINITE INTEGRALS 7 (37) Cse 8 c < <. By ssumption, f(x)dx, Hence, y The f(x)dx, nd f(x)dx exist. orem 2, c f(x)dx, f(x)dx, nd f(x)dx exist. It follows from Cse 1 tht: c (38) Tht is to sy: c c (39) nd, hence: f(x)dx f(x)dx (40) Thus, in ll cses: (41) Theorem 9. Constnts If,, c R, then cdx = c( ). Proof. First consider the cse where <. Suppose tht P : x 0 <... < x n is prtition of [, ] nd c i [x i 1, x i ], 1 i n. Then: (42) R. = Thus: (43) c( (x)) i = c ( (x)) i = c( ). lim R = lim c( ) = c( ). Secondly, consider the cse where =. Then, y definition (2), cdx = cdx = 0 nd c( ) = c( ) = c0 = 0. Hence, cdx = c( ). Thirdly, consider the cse where >. Then, y the first cse nd definition (3), cdx = cdx = c( ) = c( ). Theorem 10. Domintion If f nd g re integrle on [, ] nd f(x) g(x) for ll x [, ], then f(x)dx g(x)dx.
8 8 J. MCCARTHY Proof. Let A = f(x)dx nd B = g(x)dx, which exist since f nd g re integrle on [, ]. Let ɛ e positive rel numer. By the definition of definite integrl, there exist positive rel numers α nd β such tht if: (44) P : = x 0 <... < x m = nd: (45) Q : = y 0 <... < y n = re prtitions of [, ] with P < α nd Q < β nd: (46) c i [x i 1, x i ], 1 i m nd: (47) d j [y j 1, y j ], 1 j n, then: m (48) f(c i )( (x)) i A < ɛ nd: (49) f(d j )( (y)) j B < ɛ. j=1 Choose positive integer N such tht 1/N < α nd 1/N < β. Let m = n = N; (50) x i = + (( )i/n), 0 i N; (51) y i = + (( )i/n), 0 i N; (52) c i = x i, 1 i N; nd: (53) d i = y i, 1 i N. Note tht with these choices P = 1/N < α nd Q = 1/N < β. It follows from equtions (48) nd (49) tht: (54) A nd: N f(c i )( (x)) i < ɛ
9 DEFINITE INTEGRALS 9 N (55) g(d i )( (y)) i B < ɛ. Note tht c i = d i, 1 i N. Since f(x) g(x) for ll x [, ], it follows tht: (56) f(c i ) g(d i ), 1 i N. Note tht: (57) ( (x)) i = ( (y)) i = 1/N > 0, 1 i N. It follows from equtions (54), (55), (56), nd (57) tht: N N (58) A < f(c i )( (x)) i + ɛ g(d i )( (y)) i + ɛ < (B + ɛ) + ɛ = B + 2ɛ. This proves tht A < B + 2ɛ for every positive rel numer ɛ. Tht is to sy, A B. Theorem 11. If f(x) 0 on [, ], then f(x)dx 0. Proof. By Theorems 9 nd 10: (59) 0 = 0 ( ) = 0dx Theorem 12. MxMin Inequlity If f hs mximum vlue mxf nd minimum vlue minf on [, ], then: (60) (minf) ( ) Proof. By the definition of minf nd mxf: f(x)dx (mxf) ( ). (61) minf f(x) mxf for ll x [, ]. It follows from Theorems 9 nd 10 tht: (62) (minf) ( ) = (minf)dx f(x)dx (mxf)dx = (mxf) ( ).
10 10 J. MCCARTHY 2. Section 5.4 Theorem 13. The Men Vlue Theorem for Definite Integrls If f is continuous on [, ], then there exists c [, ] such tht: (63) f(c) = 1 Proof. Since f is continuous on [, ], it follows from the Extreme Vlue Theorem tht there exist c 1 [, ] nd c 2 [, ] such tht f(c 1 ) f(x) f(c 2 ) for ll x [, ]. It follows from Theorems 1, 9, nd 10 tht: (64) f(c 1 )( ) = f(c 1 )dx f(x)dx Since <, > 0. It follows from eqution (64) tht: (65) f(c 1 ) 1 f(x)dx f(c 2 ). f(c 2 )dx = f(c 2 )( ). Suppose, on the one hnd, tht c 1 = c 2. Then from eqution (65) we conclude tht: (66) f(c 1 ) = 1 Since c 1 [, ], the desired conclusion follows. Suppose, on the other hnd, tht c 1 c 2. Let: (67) L. = 1 It follows from equtions (65) nd (67) tht L is numer etween the vlues f(c 1 ) nd f(c 2 ) of f t the endpoints of closed intervl I contined in [, ]. Since f is continuous on [, ], f is continuous on I. It follows from the Intermedite Vlue Theorem tht there exists c I such tht: (68) f(c) = L = 1 Since c I [, ], the desired conclusion follows. Hence, in ny cse, the desired conclusion follows. Theorem 14. The Fundmentl Theorem of Clculus Prt 1 If f is continuous on [, ], then the rule: (69) F (x). = x f(t)dt defines continuous function F on [, ] which is differentile on (, ) such tht:
11 DEFINITE INTEGRALS 11 (70) F (x) = d dx x f(t)dt = f(x). Proof. Suppose tht x (, ). Let h e nonzero rel numer such tht x + h (, ). By Theorems 1 nd 8: (71) x+h f(t)dt = x By equtions (69) nd (71), it follows tht: (72) F (x + h) F (x) h f(t)dt + = 1 h x+h x x+h x f(t)dt. f(t)dt. Suppose, on the one hnd, tht h > 0. Since x, x + h [, ], [x, x + h] [, ]. Since f is continuous on [, ], it follows tht f is continuous on [x, x + h]. Thus, y Theorem 13, there exists c [x, x + h] such tht: (73) 1 h x+h x f(t)dt = f(c). Suppose, on the other hnd, tht h < 0. Since x, x + h [, ], [x + h, x] [, ]. Since f is continuous on [, ], it follows tht f is continuous on [x + h, x]. Thus, y Theorem 13, there exists c [x + h, x] such tht: (74) In other words, y Theorem 2: 1 x f(t)dt = f(c). ( h) x+h (75) Thus, in ny cse: 1 h x+h x f(t)dt = f(c). (76) 1 h x+h x f(t)dt = f(c) for some c etween x nd x + h. It follows from equtions (72) nd (76) tht: F (x + h) F (x) (77) = f(c) h for some c etween x nd x + h. Now suppose tht ɛ is positive rel numer. Since f is continuous on [, ] nd x (, ), f is continuous t x. Thus, there exists positive rel numer δ such tht if y x < δ, then f(y) f(x) < ɛ. Now suppose tht h 0 < δ. Since c is etween x nd x + h, it follows tht c x h = h 0 < δ. Hence, y the preceding prgrph, f(c) f(x) < ɛ. It follows from eqution (77) tht:
12 12 J. MCCARTHY F (x + h) F (x) (78) f(x) < ɛ. h Hence: F (x + h) F (x) (79) lim = f(x). h 0 h Tht is to sy, F (x) = f(x). This proves tht F is differentile on (, ) nd F (x) = f(x) for ll x (, ). A similr rgument shows tht the onesided derivtives of F t nd re given y the equtions F +() = f() nd F () = f(). Thus, F hs onesided derivtive t nd nd derivtive t x for ll x (, ). Since differentile functions re continuous, it follows tht F is continuous on [, ]. Theorem 15. The Fundmentl Theorem of Clculus Prt 2 If f is continuous on [, ] nd F is continuous function on [, ] such tht F is n ntiderivtive of f on (, ), then: (80) Proof. Let: F () F (). (81) G(x). = x f(t)dt for ll x [, ]. By Theorem 14, G is continuous on [, ] nd n ntiderivtive of f on (, ). Thus F nd G re ntiderivtives of f on (, ). It follows from Corollry 2 of the Men Vlue Theorem tht there exists constnt C such tht: (82) F (x) = G(x) + C for ll x (, ). Note tht F nd G re continuous on [, ] nd, hence, t nd. Thus: (83) lim F (x) = F (), x + (84) lim G(x) = G(), x + (85) lim F (x) = F (), x nd: (86) lim G(x) = G(). x
13 DEFINITE INTEGRALS 13 It follows from equtions (82), (83), (84), (85), nd (86), the sum rule for limits, nd the constnt rule for limits tht: (87) F () = lim x + F (x) = lim x + G(x) + C nd: = lim G(x) + lim C = G() + C, x + x + (88) F () = lim F (x) = lim G(x) + C x x = lim G(x) + lim C = G() + C, x x It follows from definition (2), equtions (81), (87), nd (88) tht: (89) F () F () = (G() + C) (G() + C) = G() G() = f(t)dt f(t)dt = f(t)dt 0 = f(t)dt. Since f(t)dt = f(x)dx, it follows from eqution (89) tht F () F (). Deprtment of Mthemtics, Michign Stte University, Est Lnsing, MI Emil ddress: URL:
Chapter 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 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 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 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 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 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 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 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 informationImproper Integrals. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Improper Integrls MATH 2, Clculus II J. Robert Buchnn Deprtment of Mthemtics Spring 28 Definite Integrls Theorem (Fundmentl Theorem of Clculus (Prt I)) If f is continuous on [, b] then b f (x) dx = [F(x)]
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 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 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 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 informationMath 1431 Section 6.1. f x dx, find f. Question 22: If. a. 5 b. π c. π5 d. 0 e. 5. Question 33: Choose the correct statement given that
Mth 43 Section 6 Question : If f d nd f d, find f 4 d π c π d e  Question 33: Choose the correct sttement given tht 7 f d 8 nd 7 f d3 7 c d f d3 f d f d f d e None of these Mth 43 Section 6 Are Under
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 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 informationAN INEQUALITY OF OSTROWSKI TYPE AND ITS APPLICATIONS FOR SIMPSON S RULE AND SPECIAL MEANS. I. Fedotov and S. S. Dragomir
RGMIA Reserch Report Collection, Vol., No., 999 http://sci.vu.edu.u/ rgmi AN INEQUALITY OF OSTROWSKI TYPE AND ITS APPLICATIONS FOR SIMPSON S RULE AND SPECIAL MEANS I. Fedotov nd S. S. Drgomir Astrct. An
More informationMAT137 Calculus! Lecture 27
MAT37 Clculus! Lecture 7 Tody: More out Integrls (Rest of the Videos) Antiderivtives Next: Fundmentl Theorem of Clculus NEW office hours: T & R @ BA 4 officil wesite http://uoft.me/mat37 Betriz NvrroLmed
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 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. Theorem Suppose f is continuous
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 informationCalculus of variations with fractional derivatives and fractional integrals
Anis do CNMAC v.2 ISSN 1984820X Clculus of vritions with frctionl derivtives nd frctionl integrls Ricrdo Almeid, Delfim F. M. Torres Deprtment of Mthemtics, University of Aveiro 3810193 Aveiro, Portugl
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 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 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 informationQUADRATURE is an oldfashioned word that refers to
World Acdemy of Science Engineering nd Technology Interntionl Journl of Mthemticl nd Computtionl Sciences Vol:5 No:7 011 A New Qudrture Rule Derived from Spline Interpoltion with Error Anlysis Hdi Tghvfrd
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 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 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 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 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 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 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 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 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 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 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 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 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 informationAP Calculus AB Unit 5 (Ch. 6): The Definite Integral: Day 12 Chapter 6 Review
AP Clculus AB Unit 5 (Ch. 6): The Definite Integrl: Dy Nme o Are Approximtions Riemnn Sums: LRAM, MRAM, RRAM Chpter 6 Review Trpezoidl Rule: T = h ( y + y + y +!+ y + y 0 n n) **Know how to find rectngle
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 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 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 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 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 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 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 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 informationMath 190 Chapter 5 Lecture Notes. Professor Miguel Ornelas
Mth 19 Chpter 5 Lecture Notes Professor Miguel Ornels 1 M. Ornels Mth 19 Lecture Notes Section 5.1 Section 5.1 Ares nd Distnce Definition The re A of the region S tht lies under the grph of the continuous
More informationWENJUN LIU AND QUÔ C ANH NGÔ
AN OSTROWSKIGRÜSS TYPE INEQUALITY ON TIME SCALES WENJUN LIU AND QUÔ C ANH NGÔ Astrct. In this pper we derive new inequlity of OstrowskiGrüss type on time scles nd thus unify corresponding continuous
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 informationThe Fundamental Theorem of Calculus
The Fundmentl Theorem of Clculus Professor Richrd Blecksmith richrd@mth.niu.edu Dept. of Mthemticl Sciences Northern Illinois University http://mth.niu.edu/ richrd/mth229. The Definite Integrl We define
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 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 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 informationSome estimates on the HermiteHadamard inequality through quasiconvex functions
Annls of University of Criov, Mth. Comp. Sci. Ser. Volume 3, 7, Pges 8 87 ISSN: 13693 Some estimtes on the HermiteHdmrd inequlity through qusiconvex functions Dniel Alexndru Ion Abstrct. In this pper
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 informationASYMPTOTIC BEHAVIOR OF INTERMEDIATE POINTS IN CERTAIN MEAN VALUE THEOREMS. II
STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume LV, Number 3, September 2010 ASYMPTOTIC BEHAVIOR OF INTERMEDIATE POINTS IN CERTAIN MEAN VALUE THEOREMS. II TIBERIU TRIF Dedicted to Professor Grigore Ştefn
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 information0.1 Chapters 1: Limits and continuity
1 REVIEW SHEET FOR CALCULUS 140 Some of the topics hve smple problems from previous finls indicted next to the hedings. 0.1 Chpters 1: Limits nd continuity Theorem 0.1.1 Sndwich Theorem(F 96 # 20, F 97
More informationFinal Exam  Review MATH Spring 2017
Finl Exm  Review MATH 5  Spring 7 Chpter, 3, nd Sections 5.5.5, 5.7 Finl Exm: Tuesdy 5/9, :37:pm The following is list of importnt concepts from the sections which were not covered by Midterm Exm or.
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 informationMath 120 Answers for Homework 13
Mth 12 Answers for Homework 13 1. In this problem we will use the fct tht if m f(x M on n intervl [, b] (nd if f is integrble on [, b] then (* m(b f dx M(b. ( The function f(x = 1 + x 3 is n incresing
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 informationMAT137 Calculus! Lecture 28
officil wesite http://uoft.me/mat137 MAT137 Clculus! Lecture 28 Tody: Antiderivtives Fundmentl Theorem of Clculus Net: More FTC (review v. 8.58.7) 5.7 Sustitution (v. 9.19.4) Properties of the Definite
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 informationTest 3 Review. Jiwen He. I will replace your lowest test score with the percentage grade from the final exam (provided it is higher).
Test 3 Review Jiwen He Test 3 Test 3: Dec. 46 in CASA Mteril  Through 6.3. No Homework (Thnksgiving) No homework this week! Hve GREAT Thnksgiving! Finl Exm Finl Exm: Dec. 1417 in CASA You Might Be Interested
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 informationSection Areas and Distances. Example 1: Suppose a car travels at a constant 50 miles per hour for 2 hours. What is the total distance traveled?
Section 5.  Ares nd Distnces Exmple : Suppose cr trvels t constnt 5 miles per hour for 2 hours. Wht is the totl distnce trveled? Exmple 2: Suppose cr trvels 75 miles per hour for the first hour, 7 miles
More informationSOME INTEGRAL INEQUALITIES OF GRÜSS TYPE
RGMIA Reserch Report Collection, Vol., No., 998 http://sci.vut.edu.u/ rgmi SOME INTEGRAL INEQUALITIES OF GRÜSS TYPE S.S. DRAGOMIR Astrct. Some clssicl nd new integrl inequlities of Grüss type re presented.
More informationMath 3B: Lecture 9. Noah White. October 18, 2017
Mth 3B: Lecture 9 Noh White October 18, 2017 The definite integrl Defintion The definite integrl of function f (x) is defined to be where x = b n. f (x) dx = lim n x n f ( + k x) k=1 Properties of definite
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 informationBig idea in Calculus: approximation
Big ide in Clculus: pproximtion Derivtive: f (x) = df dx f f(x +h) f(x) =, x h rte of chnge is pproximtely the rtio of chnges in the function vlue nd in the vrible in very short time Liner pproximtion:
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 informationThe presentation of a new type of quantum calculus
DOI.55/tmj2722 The presenttion of new type of quntum clculus Abdolli Nemty nd Mehdi Tourni b Deprtment of Mthemtics, University of Mzndrn, Bbolsr, Irn Emil: nmty@umz.c.ir, mehdi.tourni@gmil.com b Abstrct
More informationChapters 4 & 5 Integrals & Applications
Contents Chpters 4 & 5 Integrls & Applictions Motivtion to Chpters 4 & 5 2 Chpter 4 3 Ares nd Distnces 3. VIDEO  Ares Under Functions............................................ 3.2 VIDEO  Applictions
More informationSection 5.4 Fundamental Theorem of Calculus 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus 1
Section 5.4 Fundmentl Theorem of Clculus 2 Lectures College of Science MATHS : Clculus (University of Bhrin) Integrls / 24 Definite Integrl Recll: The integrl is used to find re under the curve over n
More informationChapter 6 Notes, Larson/Hostetler 3e
Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn
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 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 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 informationHandout I  Mathematics II
Hndout I  Mthemtics II The im of this hndout is to briefly summrize the most importnt definitions nd theorems, nd to provide some smple exercises. The topics re discussed in detil t the lectures nd seminrs.
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 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 informationChapter 8.2: The Integral
Chpter 8.: The Integrl You cn think of Clculus s doulewide triler. In one width of it lives differentil clculus. In the other hlf lives wht is clled integrl clculus. We hve lredy eplored few rooms in
More information(4.1) D r v(t) ω(t, v(t))
1.4. Differentil inequlities. Let D r denote the right hnd derivtive of function. If ω(t, u) is sclr function of the sclrs t, u in some open connected set Ω, we sy tht function v(t), t < b, is solution
More informationThe Evaluation Theorem
These notes closely follow the presenttion of the mteril given in Jmes Stewrt s textook Clculus, Concepts nd Contexts (2nd edition) These notes re intended primrily for inclss presenttion nd should not
More informationT b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions.
Rel Vribles, Fll 2014 Problem set 5 Solution suggestions Exerise 1. Let f be bsolutely ontinuous on [, b] Show tht nd T b (f) P b (f) f (x) dx [f ] +. Conlude tht if f is in AC then it is the differene
More informationGENERALIZATIONS OF WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS AND ITS APPLICATIONS. (b a)3 [f(a) + f(b)] f x (a,b)
GENERALIZATIONS OF WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS AND ITS APPLICATIONS KUEILIN TSENG, GOUSHENG YANG, AND SEVER S. DRAGOMIR Abstrct. In this pper, we estblish some generliztions
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 informationUnit Six AP Calculus Unit 6 Review Definite Integrals. Name Period Date NONCALCULATOR SECTION
Unit Six AP Clculus Unit 6 Review Definite Integrls Nme Period Dte NONCALCULATOR SECTION Voculry: Directions Define ech word nd give n exmple. 1. Definite Integrl. Men Vlue Theorem (for definite integrls)
More informationDefinition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim
Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)
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 informationRiemann Integrals and the Fundamental Theorem of Calculus
Riemnn Integrls nd the Fundmentl Theorem of Clculus Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University September 16, 2013 Outline Grphing Riemnn Sums
More informationMA 124 January 18, Derivatives are. Integrals are.
MA 124 Jnury 18, 2018 Prof PB s oneminute introduction to clculus Derivtives re. Integrls re. In Clculus 1, we lern limits, derivtives, some pplictions of derivtives, indefinite integrls, definite integrls,
More informationMath& 152 Section Integration by Parts
Mth& 5 Section 7.  Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible
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 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 information