The First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).


 Melissa Casey
 4 years ago
 Views:
Transcription
1 The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples of these theorems in this set of notes. The net set of notes will consider some pplictions of these theorems. The First Fundmentl Theorem of Clculus. If f() is continuous on [, b nd F () is ny ntiderivtive of f() on [, b, then f() d = F (b) F (). In words: In order to compute definite integrl f(), it suffices to find n ntiderivtive of f(), then compute the difference of this ntiderivtive. The Second Fundmentl Theorem of Clculus. If f() is continuous on n intervl nd is ny number in this intervl, then the function is n ntiderivtive of f(). In other words, A() = d d. In words: In order to constuct n ntiderivtive of function f(), it is enough to be ble to compute the definite integrl of f(). Both of these theorems estblish reltionship between ntiderivtives (i.e., indefinite integrls) nd net res (i.e., definite integrls.) In effect, the First Fundmentl Theorem sys tht definite integrtion reverses differentition, wheres the Second Fundmentl Theorem sys tht differentition reverses definite integrtion. Thus, the Fundmentl Theorems sy tht definite integrtion is ctully sort of ntidifferentition. This eplins why we use integrtion nottion nd terminology (indefinite integrls) for ntiderivtives. From mthemticl point of view, the Second Fundmentl Theorem is the true fundmentl theorem of clculus. The First Fundmentl Theorem is just logicl consequence of Second Fundmentl Theorem. However, the First Fundmentl Theorem is wht most nonmthemticins think of s the fundmentl theorem. This is becuse the First Fundmentl Theorem is the one tht ppers in pplictions in lots of different subjects. Thus, the First Fundmentl Theorem is of prcticl interest wheres the Second Fundmentl Theorem is primrily of theoreticl interest, lthough it does hve some prcticl pplictions. Emple : Find. Using the First Fundmentl Theorem of Clculus sin θ dθ. Solution: We lredy know this integrl is 0 by the previous set of notes. But lets compute this using the First Fundmentl Theorem. First, we need to find n ntiderivtive of sin θ. But sin θ dθ = cos θ + C,
2 so we ll use F (θ) = cos θ in the fundmentl theorem. (The fundmentl theorem only requires tht we find some ntiderivtive, not the most generl ntiderivtive, so we don t need to use the +C.) So, sin θ dθ = F ( 3 ) F ( ) = cos (3 ) ( cos ( )) = = 0, consistent with the nswer we obtined in the lst set of notes. We mke two comments on the bove emple. First, wht if we used different ntiderivtive? Could tht hve chnged our nswer? Fortuntely the nswer is no. The most generl ntiderivtive is F (θ) = cos θ + C, so if we used this we would hve hd sin θ dθ = F ( 3 ) F ( ) = ( cos (3 ) + C) ( cos ( ) + C) = 0 + C + 0 C = 0. The C s will lwys cncel off like this, so we don t need the +C when computing definite integrls. Second, we will find it convenient to write Thus, we could write Emple : Find Solution: First we find n ntiderivtive: d = We ll use s our ntiderivtive. Thus Emple 3: Find 4 F () b = F (b) F (). sin θ dθ = F (θ) 3 4 d. =.... / d = / + C. d = 4 = 4 =. ( 3 + 5) d. Solution: Often times, we will just compute the ntiderivtives in our heds. Thus ( 3 + 5) d = ( ) = ( ) ( 4 ( )4 + 5 ( ) ) =.5 Emple 4: Things cn go wrong if the integrnd hs verticl symptote. (Plese note the Fundmentl Theorem does require f() to be defined nd continuous on the whole intervl of integrtion.) For emple, the following clcultion is wrong. d = = () ( ( ) ) =. We cn tell immeditely tht something is wrong since d is the net re of figure tht is lwys bove the is, nd therefore the integrl better be postive. Furthermore, using pproimting sums, it looks s though the integrl is infinite. Thus, while the fundmentl theorem gives us very powerful
3 method to compute definite integrls, we do need to be creful nd check tht our function stisfies the hypotheses of the fundmentl theorem before using the conclusion of tht theorem. Hving sid tht, I should point out tht, lthough s stted the fundmentl theorem requires the function to be continuous, in ctulity the fundmentl theorem holds for clss of functions more generl thn the continuous functions. In prticulr, if the integrnd hs finite number of jump discontinuities then the Fundmentl Theorem still works. Integrls whose integrnds contin symptotes re referred to s improper integrls nd re studied in detil in Clculus. A lot of integrls tht rise in prctice (especilly in Probbility Theory nd Quntum Mechnics) re improper integrls.. Using the Second Fundmentl Theorem of Clculus A lrge prt of the difficulty in understnding the Second Fundmentl Theorem of Clculus is getting grsp on the function. As definite integrl, we should think of A() s giving the net re of geometric figure. The function A() depends on three different things. First, it depends on the integrnd f(t), different integrnd gives rise to different A(). In terms of res, the integrnd determines the height of the figure. Net, A() depends on. tells us where the firgure strts. Finlly, A() depends on, n tells us where the figure stops. Now, we usully think of f(t) nd s being fied, n is our vrible. For emple, if f(t) = t nd =, then A() gives us the re of trpezoid, but the bse of this trpezoid is not necessrily given. See the figures below. Thus, A() = 0 since the trpezpoid degenertes into verticl line. A() is the re of trpezoid with bse = nd heights nd, thus its re is ( ) + = 3. Generlly, A() is trpezoid with bse nd heights n, so A() = ( ) + Now we cn esily compute the derivtive of A(): A () = =. =. This required quite bit of work. First, we hd to write down formul for A(), then we hd to tke the derivtive. The Second Fundmentl Theorem tells us tht we didn t ctully need to find n eplicit formul for A(), tht we could immeditely write down A () =. We remind ourselves of the Second Fundmentl Theorem. The Second Fundmentl Theorem of Clculus. If f() is continuous on n intervl nd is ny number in tht intervl, then = f(). d In words: Integrting f(t) up to, then computing the derivtive of this with respect to is the sme s evluting the integrnd t, i.e., the sme s plugging into f(t). We turn to some more emples. 3 Emple 5: Find (t + 3t) dt. d Solution: By the second Fundmentl Theorem of Clculus, we need only evlute the integrnd, t + 3t, t. Hence (t + 3t) dt = + 3. d
4 4 (You could just s well hve first computed the definite integrl using the first Fundmentl Theorem, then computed the derivtive. Doing this should gin give + 3. But doing so misses the whole point of the Second Fundmentl Theorem.) Emple 6: Find sin (θ ) dθ. d 0 Solution: A few notes bck we mentioned tht sin (θ ) hs no ntiderivtive in terms of simple functions, so it is impossible to compute this by first computing the definite integrl, then computing the derivtive. But this doesn t bother us since we know the Second Fundmentl Theorem: sin (θ ) dθ = sin ( ). d 0 Mtters re bit more complicted if the limits of integrtion re not simply. The most generl sitution llows both the upper nd lower limits of integrtion to be functions of. The Second Fundmentl Theorem of Clculus with rbitrry limits of integrtion. If f() is continuous nd u() nd l() re differentible functions, then [ d u() = u ()f(u()) l ()f(l()). d l() Emple 7: According to this form of the Fundmentl Theorem, e t dt = () e ( ) e = e + e. d This hs nice geometric interprettion. As increses, the re represented by et dt epnds both to the left nd to the right. On the right we dd little rectngle of height e, on the left we dd little rectngle of height e. Emple 8: Emple 9: Emple 0: Let G() = ln 0 Solution: [ cos t dt = ( ) cos ( ) ( ) cos ( ) = cos ( ), d d e t 3 dt = (e ) (e ) 3 (3) (3) 3 = e d 3 e t / dt. Find G (). G () = d ln e t / dt = (ln ) e (ln ) / = d 0 e (ln ) /. The lower limit of integrtion is constnt, so the (0) e 0 / term is just 0, so we did not include tht term. We ll see lots of pplictions of the First Fundmentl Theorem of Clculus in the net set of notes. Direct pplictions of the Second Fundmentl Theorem re bit hrder to come by, but pplictions do eist for it. In prticulr, the lst emple is typicl sort of clcultion you might do in probbility clss (specificlly, this emple reltes the probbility densities of the norml distribution nd the lognorml distribution.)
5 3. Theory We look t some of the more theoreticl spects of the Fundmentl Theorems in this section. We begin by outlining proof of the fundmentl theorems. Actully, this rgument should be very fmilir to you by now. We ve gone through this rgument in severl specific cses lredy. Even though I m tking the time to go through the proofs of these theorems, you will not be responsible for either of these proofs. Being ble to do clculutions like the emples t the beginning of the set of notes on Ares nd Antiderivtives is fr more importnt thn memorizing this proof. After you ve done enough emples like those you should see tht the proof of the fundmentl theorem is the ect sme rgument just written out in bstrct generlity. We begin with the Second Fundmentl Theorem: We suppose f() is continuous on n intervl, is in the intervl, nd we need to show tht the function A() = is n ntiderivtive of f(t). In other words, we need to show tht A () = f(), or = f(). d We don t hve n eplicit formul for the integrl, so our only hope for computing the derivtive is to write out the definition of derivtive: +h = lim. d The numertor is the difference between two net res, the first from to + h, the second from to. Thus, the difference is the net re from to + h, i.e., +h = +h. If h is very smll then this lst integrl is the net re of figure tht is essentilly rectngle with width h nd height f(), so +h Piecing together everything, = lim d = lim f() h. +h +h f() h lim = f(). This only shows f(), d so techniclly this is not proof of the second Fundmentl Theorem. To get rigorous proof we would need to quntify just how good the pproimtion +h f() h is. Requiring f() to be continuous is enough to gurntee tht this pproimtion is good enough. You should look bck t the first two emples in the the notes on Ares nd Antiderivtives. The proof of the Second Fundmentl Theorem of Clculus is ectly the sme rgument we used in those specific emples. 5
6 6 Now for the first Fundmentl Theorem. Honestly, the proof I m going to give is bit confusing the first time you see it. The problem is tht the First Fundmentl Theorem of Clculus is relly just sying the sme thing s the Second Fundmentl Theorem, but just in different wy, so it my pper tht we re not doing much in the proof. (There ctully is something going on. To go from the Second to the First, we need to pply the +C Theorem, nd the +C Theorem is consequnce of the Men Vlue Theorem) Thus, we ssume f() is continuous on [, b nd we let F () be n rbitrry ntiderivtive of f(). We need to show = F (b) F (). (I ve swithced the vrible of integrtion from n to t only becuse I wnt to use for something else.) Do we know n ntiderivtive of f()? Yes, the Second Fundmentl Theorem tells us tht A() = is n ntiderivtive of f(). Furthermore, since both F () nd A() re ntiderivtives of f() then there is constnt C so tht F () = A() + C. Also, A(b) A() = = ; = 0 since this integrl is the net re of rectngle of height f() nd width 0. But Therefore F (b) F () = (A(b) + C) (A() + C) = (A(b) A()) + (C C) = A(b) A(). F (b) F () = where F () is ny ntiderivtive of f() on [, b. It is possible to prove the First Fundmentl Theorem of Clculus directly, tht is, without recourse to the Second Fundmentl Theorem. Such proof requires doing creful estimtes with Riemnn sums. The dvntge of the bove proof of the Fundmentl Theorems is tht we did not hve to del directly with the Riemnn Sums. We close by writting down ech of the Fundmentl Theorems in yet nother form. The First Fundmentl Theorem of Clculus. If F () is continuous on [, b then F () d = F (b) F (). The originl sttement involved two functions, f() nd F (), nd F () ws n ntiderivtive of f(). But this just mens tht F () = f(), so we replce f() with F (). This resttement fits in nicely with the Second Fundmentl Theorem. Wheres the Second Fundmentl Theorem sys tht tking the derivtive of n integrl brings you bck to the originl function, this form of the First Fundmentl Theorem sys tht tking the integrl of derivtive brings you bck to the originl function. The Second Fundmentl Theorem of Clculus. If f() is continuous on n intervl contining, then +h lim = f(). The originl sttement sys d d = +h using +h = f(). But writting out the definition of the derivtive then gives this lterntive form. Corollry: If f() is continous on [, b then f() hs n ntiderivtive. This follows directly from the Second Fundmentl Theorem. ntiderivtive of f(). Tht theorem eplicitly describes n
and that at t = 0 the object is at position 5. Find the position of the object at t = 2.
7.2 The Fundmentl Theorem of Clculus 49 re mny, mny problems tht pper much different on the surfce but tht turn out to be the sme s these problems, in the sense tht when we try to pproimte solutions we
More 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 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 informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More 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 informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More 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 informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationUNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3
UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,
More informationSYDE 112, LECTURES 3 & 4: The Fundamental Theorem of Calculus
SYDE 112, LECTURES & 4: The Fundmentl Theorem of Clculus So fr we hve introduced two new concepts in this course: ntidifferentition nd Riemnn sums. It turns out tht these quntities re relted, but it is
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums  1 Riemnn
More 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 informationChapter 0. What is the Lebesgue integral about?
Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous
More 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 informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 7.. Introduction to Integrtion. 7. Integrl Clculus As ws the cse with the chpter on differentil
More 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 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 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 information5.7 Improper Integrals
458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the
More 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 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 informationReview of basic calculus
Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below
More 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 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 informationAP Calculus Multiple Choice: BC Edition Solutions
AP Clculus Multiple Choice: BC Edition Solutions J. Slon Mrch 8, 04 ) 0 dx ( x) is A) B) C) D) E) Divergent This function inside the integrl hs verticl symptotes t x =, nd the integrl bounds contin this
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 informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationOverview of Calculus I
Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,
More 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 informationMath 131. Numerical Integration Larson Section 4.6
Mth. Numericl Integrtion Lrson Section. This section looks t couple of methods for pproimting definite integrls numericlly. The gol is to get good pproimtion of the definite integrl in problems where n
More informationThe practical version
Roerto s Notes on Integrl Clculus Chpter 4: Definite integrls nd the FTC Section 7 The Fundmentl Theorem of Clculus: The prcticl version Wht you need to know lredy: The theoreticl version of the FTC. Wht
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 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 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 informationp(t) dt + i 1 re it ireit dt =
Note: This mteril is contined in Kreyszig, Chpter 13. Complex integrtion We will define integrls of complex functions long curves in C. (This is bit similr to [relvlued] line integrls P dx + Q dy in R2.)
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More 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 informationMain topics for the Second Midterm
Min topics for the Second Midterm The Midterm will cover Sections 5.45.9, Sections 6.16.3, nd Sections 7.17.7 (essentilly ll of the mteril covered in clss from the First Midterm). Be sure to know the
More informationdifferent methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s).
Mth 1A with Professor Stnkov Worksheet, Discussion #41; Wednesdy, 12/6/217 GSI nme: Roy Zho Problems 1. Write the integrl 3 dx s limit of Riemnn sums. Write it using 2 intervls using the 1 x different
More 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 informationIntegration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
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 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 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 information1 The Riemann Integral
The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re
More informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
More information4.4 Areas, Integrals and Antiderivatives
. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order
More information5.3 The Fundamental Theorem of Calculus
CHAPTER 5. THE DEFINITE INTEGRAL 35 5.3 The Funmentl Theorem of Clculus Emple. Let f(t) t +. () Fin the re of the region below f(t), bove the tis, n between t n t. (You my wnt to look up the re formul
More informationHow can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function?
Mth 125 Summry Here re some thoughts I ws hving while considering wht to put on the first midterm. The core of your studying should be the ssigned homework problems: mke sure you relly understnd those
More informationStuff You Need to Know From Calculus
Stuff You Need to Know From Clculus For the first time in the semester, the stuff we re doing is finlly going to look like clculus (with vector slnt, of course). This mens tht in order to succeed, you
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 informationInterpreting Integrals and the Fundamental Theorem
Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of
More informationBob Brown Math 251 Calculus 1 Chapter 5, Section 4 1 CCBC Dundalk
Bo Brown Mth Clculus Chpter, Section CCBC Dundlk The Fundmentl Theorem of Clculus Informlly, the Fundmentl Theorem of Clculus (FTC) sttes tht differentition nd definite integrtion re inverse opertions
More information1 Functions Defined in Terms of Integrals
November 5, 8 MAT86 Week 3 Justin Ko Functions Defined in Terms of Integrls Integrls llow us to define new functions in terms of the bsic functions introduced in Week. Given continuous function f(), consider
More informationWeek 10: Line Integrals
Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.
More 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 informationTopics Covered AP Calculus AB
Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.
More information2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).
AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following
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 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 informationMath Calculus with Analytic Geometry II
orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove xxis) ( bove f under xxis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem
More informationa < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1
Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the
More informationSection 4: Integration ECO4112F 2011
Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic
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 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 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 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 informationCMDA 4604: Intermediate Topics in Mathematical Modeling Lecture 19: Interpolation and Quadrature
CMDA 4604: Intermedite Topics in Mthemticl Modeling Lecture 19: Interpoltion nd Qudrture In this lecture we mke brief diversion into the res of interpoltion nd qudrture. Given function f C[, b], we sy
More informationF (x) dx = F (x)+c = u + C = du,
35. The Substitution Rule An indefinite integrl of the derivtive F (x) is the function F (x) itself. Let u = F (x), where u is new vrible defined s differentible function of x. Consider the differentil
More informationChapter 6 Techniques of Integration
MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln
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 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 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 informationMATH 144: Business Calculus Final Review
MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationContinuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom
Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive
More informationNow, given the derivative, can we find the function back? Can we antidifferenitate it?
Fundmentl Theorem of Clculus. Prt I Connection between integrtion nd differentition. Tody we will discuss reltionship between two mjor concepts of Clculus: integrtion nd differentition. We will show tht
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 informationReversing the Chain Rule. As we have seen from the Second Fundamental Theorem ( 4.3), the easiest way to evaluate an integral b
Mth 32 Substitution Method Stewrt 4.5 Reversing the Chin Rule. As we hve seen from the Second Fundmentl Theorem ( 4.3), the esiest wy to evlute n integrl b f(x) dx is to find n ntiderivtive, the indefinite
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 informationAppendix 3, Rises and runs, slopes and sums: tools from calculus
Appendi 3, Rises nd runs, slopes nd sums: tools from clculus Sometimes we will wnt to eplore how quntity chnges s condition is vried. Clculus ws invented to do just this. We certinly do not need the full
More informationChapter 8: Methods of Integration
Chpter 8: Methods of Integrtion Bsic Integrls 8. Note: We hve the following list of Bsic Integrls p p+ + c, for p sec tn + c p + ln + c sec tn sec + c e e + c tn ln sec + c ln + c sec ln sec + tn + c ln
More informationChapter 5. Numerical Integration
Chpter 5. Numericl Integrtion These re just summries of the lecture notes, nd few detils re included. Most of wht we include here is to be found in more detil in Anton. 5. Remrk. There re two topics with
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 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 informationImproper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.
Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:
More informationThe Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.
Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F
More information7. Indefinite Integrals
7. Indefinite Integrls These lecture notes present my interprettion of Ruth Lwrence s lecture notes (in Herew) 7. Prolem sttement By the fundmentl theorem of clculus, to clculte n integrl we need to find
More informationLine and Surface Integrals: An Intuitive Understanding
Line nd Surfce Integrls: An Intuitive Understnding Joseph Breen Introduction Multivrible clculus is ll bout bstrcting the ides of differentition nd integrtion from the fmilir single vrible cse to tht of
More 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 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 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 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 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 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 informationMath 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8
Mth 3 Fll 0 The scope of the finl exm will include: Finl Exm Review. Integrls Chpter 5 including sections 5. 5.7, 5.0. Applictions of Integrtion Chpter 6 including sections 6. 6.5 nd section 6.8 3. Infinite
More informationRecitation 3: More Applications of the Derivative
Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech
More 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 informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
More information. Doubleangle formulas. Your answer should involve trig functions of θ, and not of 2θ. cos(2θ) = sin(2θ) =.
Review of some needed Trig Identities for Integrtion Your nswers should be n ngle in RADIANS rccos( 1 2 ) = rccos(  1 2 ) = rcsin( 1 2 ) = rcsin(  1 2 ) = Cn you do similr problems? Review of Bsic Concepts
More information