# Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

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1 Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1

2 Riemnn Sums - 1 Riemnn Sums Are estimtions like LEF T (n) nd RIGHT (n) re often clled Riemnn sums, fter the mthemticin Bernhrd Riemnn ( ) who formlized mny of the techniques of clculus. The generl form for Riemnn Sum is f(x 1 ) x + f(x 2 ) x f(x n) x n = f(x i ) x where ech x i is some point in the intervl [x i 1, x i ]. For LEF T (n), we choose the left hnd endpoint of the intervl, so x i = x i 1; for RIGHT (n), we choose the right hnd endpoint, so x i = x i. i=1 x b x 1 x 2 x 3... x n 1 0 x n x

3 Riemnn Sums - 2 The common property of ll these pproximtions is tht they involve sum of rectngulr res, with widths ( x), nd heights (f(x i )) There re other Riemnn Sums tht give slightly better estimtes of the re underneth grph, but they often require extr computtion. We will exmine some of these other clcultions little lter.

4 The Definite Integrl - 1 The Definite Integrl We observed tht s we increse the number of rectngles used to pproximte the re under curve, our estimte of the re under the grph becomes more ccurte. This implies tht if we wnt to clculte the exct re, we would wnt to use limit. The re underneth the grph of f(x) between x = nd x = b is n equl to lim LEF T (n) = lim f(x n n i 1 ) x, where x = b n. i=1

5 The Definite Integrl - 2 b b b b x x x x

6 The Definite Integrl - 3 This limit is clled the definite integrl of f(x) from to b, nd is equl to the re under curve whenever f(x) is non-negtive continuous function. The definite integrl is written with some specil nottion.

7 The Definite Integrl - 4 Nottion for the Definite Integrl The definite integrl of f(x) between x = nd x = b is denoted by the symbol f(x) dx We cll nd b the limits of integrtion nd f(x) the integrnd. The dx denotes which vrible we re using; this will become importnt for using some techniques for clculting definite integrls. Note tht this nottion shres the sme common structure with Riemnn sums: sum ( sign) widths (dx), nd heights (f(x))

8 The Definite Integrl - 5 Problem. Write the definite integrl representing the re underneth the grph of f(x) = x + cos x between x = 2 nd x = 4.

9 The Definite Integrl nd LEFT vs RIGHT - 1 The Definite Integrl - LEF T (n) vs RIGHT (n) s n We might be concerned tht we defined the re nd the definite integrl using the left hnd sum. Would we get the sme nswer for the definite integrl if we used the right hnd sum, or ny other Riemnn sum? In fct, the limit using ny Riemnn sum should give us the sme nswer. Let us look t the left nd right hnd sums for the function 2 x on the intervl from x = 1 to x = 3. Problem. Clculte LEF T (2) RIGHT (2) for x dx. Tht is, how big is the difference between these two estimtes of the re under y = 2 x over x = ?

10 The Definite Integrl nd LEFT vs RIGHT - 2 Clculte LEF T (4) RIGHT (4) for x dx.

11 Clculte LEF T (n) RIGHT (n) for 3 1 The Definite Integrl nd LEFT vs RIGHT x dx.

12 The Definite Integrl nd LEFT vs RIGHT - 4 Wht will the limit of this LEF T (n) RIGHT (n) difference be s n? Wht does this tell us bout wht would hppen if we defined the definite integrl in terms of the right hnd sum? f(x) dx = lim LEF T (n) vs. n lim RIGHT (n)? n

13 Negtive Integrl Vlues - 1 Negtive Integrl Vlues So fr we hve only delt with positive functions. Will the definite integrl still be equl to the re underneth the grph if f(x) is lwys negtive? Wht hppens if f(x) crosses the x-xis severl times? Problem. Suppose tht f(t) hs the grph shown below, nd tht A, B, C, D, nd E re the res of the regions shown. If we were to prtition [, b] into smll subintervls nd construct corresponding Riemnn sum, then the first few terms in the Riemnn sum would correspond to the region with re A, the next few to B, etc.

14 Negtive Integrl Vlues - 2 Which of these sets of terms hve positive vlues? Which of these sets hve negtive vlues?

15 Negtive Integrl Vlues - 3 Problem. Express the integrl res A, B, C, D, nd E. () f(t) dt = A + B + C + D + E f(t) dt in terms of the (positive) (b) (c) (d) f(t) dt = A - B + C - D + E f(t) dt = -A + B - C + D - E f(t) dt = -A - B - C - D - E

16 Negtive Integrl Vlues - 4 If f(t) represents velocity, wht do the negtive res in B nd D represent? () The res B nd D represent negtive positions. (b) The res B nd D represent bckwrds motion. (c) The res B nd D represent distnce trvelled bckwrds.

17 Negtive Integrl Vlues - 5 The Role of Riemnn sums (1) Riemnn sums re needed to sy wht we men by n integrl. (2) Riemnn sums enble us to decide which integrl is pproprite in word problem. (3) Riemnn sums cn lso be used to give n pproximte vlue of the integrl.

18 The Fundmentl Theorem of Clculus - Theory - 1 The Fundmentl Theorem of Clculus We hve now drwn firm reltionship between our Riemnn sums nd re clcultions (nd the physicl properties tht cn be tied to n re clcultion on grph). The time hs now come to build method to compute these res in systemtic wy. The Fundmentl Theorem of Clculus If f is continuous on the intervl [, b], nd we define relted function F (x) such tht F (x) = f(x), then f(x) dx = F (b) F ()

19 The Fundmentl Theorem of Clculus - Theory - 2 The fundmentl theorem ties: the re clcultion of definite integrl bck to our erlier slope clcultions from derivtives. However, it chnges the direction in which we tke the derivtive: Given f(x), we find the slope by finding the derivtive of f(x), or f (x). Given f(x), we find the re defined by f(x) dx by finding F (x) which is the nti-derivtive of f(x); i.e. function F (x) for which F (x) = f(x). In other words, if we cn find n nti-derivtive F (x), then clculting the vlue of the definite integrl requires simple evlution of F (x) t two points (F (b) F ()). This lst step is much esier thn computing n re using finite Riemnn sums, nd lso provides n exct vlue of the integrl insted of n estimte.

20 The Fundmentl Theorem of Clculus - Exmple - 1 Problem. Use the Fundmentl Theorem of Clculus to find the re bounded by the x-xis, the line x = 2, nd the grph y = x 2. Use the fct tht d ( ) 1 dx 3 x3 = x 2.

21 The Fundmentl Theorem of Clculus - Exmple - 2 Problem. We used the fct tht F (x) = 1 3 x3 is n nti-derivtive of x 2, so we were ble use the Fundmentl Theorem. Give nother function F (x) which would lso stisfy d dx F (x) = x2. Use the Fundmentl Theorem gin with this new function to find the re implied by 2 0 x 2 dx.

22 The Fundmentl Theorem of Clculus - Exmple - 3 Did the re/definite integrl vlue chnge? Why or why not? Bsed on tht result, give the most generl version of F (x) you cn think of. Confirm tht d dx F (x) = x2.

23 Bsic Anti-Derivtives - Reference - 1 With our extensive prctice with derivtives erlier, we should find it strightforwrd to determine some simple nti-derivtives. Complete the following tble of nti-derivtives. function f(x) nti-derivtive F(x) x 2 x C x n x 2 + 3x 2

24 Bsic Anti-Derivtives - Reference - 2 function f(x) nti-derivtive F(x) cos x sin x x + sin x

25 function f(x) nti-derivtive F(x) Bsic Anti-Derivtives - Reference - 3 e x 2 x 1 1 x x 2 1 x

26 The Specil Cse of 1/x - 1 The Specil Cse of f(x) = 1 x Problem. Sketch the grphs of f(x) = 1 x nd F (x) = ln(x).

27 The Specil Cse of 1/x - 2 A technicl fct bout integrls is tht if f(x) is continuous on its domin, it must hve n nti-derivtive on tht domin. How is tht seemingly violted by the pir f(x) = 1 x nd F (x) = ln(x)?

28 The Specil Cse of 1/x - 3 Show tht F (x) = ln( x ) stisfies F (x) = 1/x for both positive nd negtive x vlues. in the erlier tble if neces- Correct your nti-derivtive entry for 1 x sry.

29 Anti-Derivtive Prctice - 1 Problem. Find the most generl nti-derivtive of f(x) = x3 3x + 5. x

30 Anti-Derivtive Prctice - 2 Problem. Suppose we wnt to clculte the re under one section of the grph of sin x, the prt from 0 to π. f(x) = sin(x) 1 Then we should clculte A. cos(π) 0 π 2 π B. cos(0) cos(π) C. cos(π) cos(0) D. sin(π) sin(0)

31 Problem. Clculte 4 0 x dx. Anti-Derivtive Prctice - 3

32 Anti-Derivtive Prctice - 4 Nottion The expression F (b) F () comes up so often tht there is specil nottion for it. It is written s b F (x) or [F (x)] b Problem. Clculte π/3 π/4 3 sec 2 (θ) dθ.

33 Properties of Integrls - 1 Properties of Integrls Before we go on to refine our skill t clculting integrls, we should first reflect on some bsic properties of integrls tht derive from their origins s limits of more nd more ccurte Riemnn sums.

34 Properties of Integrls - 2 (1) If > b then (2) If = b then (3) (4) (5) (6) c dx = c (b ). f(x) dx = f(x) dx = 0. [ f(x) ± g(x) ] dx = c f(x) dx = c f(x) dx = c f(x) dx. f(x) dx + b f(x) dx ± c f(x) dx. f(x) dx. g(x) dx.

35 Properties of Integrls - 3 Properties 3-6 need proofs; they depend on the nlogous properties for Riemnn Sums. See Section 5.2 in the textbook for creful proofs of ll of these properties, s well s for properties 7-10 ppering lter.

36 Properties of Integrls - 4 Problem. Prove Property 4 below using the definition of the integrl. [ ] b f(x) g(x) dx = f(x) dx g(x) dx

37 Problem. Consider function for which 2 0 f(x) dx = 3, Use Property 6, 4 1 determine the vlue of f(x) dx = 2 nd f(x) dx = 2 1 c 4 0 f(x) dx + f(x) dx = 4. c Properties of Integrls - 5 f(x) dx, to help f(x) dx. A sketch might help you.

38 Properties of Integrls - 6 Other Properties of Integrls (7) If f(x) 0 for x b then (8) If f(x) g(x) for x b then (9) If m f(x) M for x b then (10) m (b ) f(x) dx f(x) dx M (b ). f(x) dx. f(x) dx 0. f(x) dx g(x) dx.

39 Properties of Integrls - 7 The following digrms illustrte the ides of Properties 8 nd 9, (8) If f(x) g(x) for x b then f(x) dx (9) If m f(x) M for x b then m (b ) M (b ). M f f b g(x) dx. f(x) dx g m b In the second digrm, it cn be observed tht the re under f(x) in [, b ] is greter thn the re under m in [, b ] but less thn the re under M. b

40 Properties of Integrls - 8 Problem. Let f(x) be s shown A C E B D b Sketch f(x) nd so demonstrte one instnce of Property 10, f(x) dx f(x) dx.

41 Net Chnge Theorem - 1 Net Chnge Theorem Note tht we crete n nti-derivtive F (x), we re building it such tht f(x) = F (x). This mens tht f gives the rte of chnge of F. Notice tht this observtion ws mde much erlier, when we strted our discussion of integrtion: when n integrl is ssocited with process of ccumultion then the rte of ccumultion is lwys precisely the integrnd.

42 Net Chnge Theorem - 2 Consider F (x) s the quntity we re trcking, so F is its rte of chnge. Another sttement of the Fundmentl Theorem of Clculus Prt 2 would then be F (x) dx = F (b) F (). The integrl of rte of chnge is the totl chnge. The textbook clls this the Net Chnge Theorem.

43 Net Chnge Theorem - 3 Problem. If cr is moving t v(t) = 2t m/s from t = 1 to t = 5, wht does the quntity 5 1 v(t) dt represent? A. The position of the cr t t = 5, strting t t = 1. B. The net chnge in position of the cr between t = 1 nd t = 5. C. The velocity of the cr t t = 5, strting t t = 1. D. The net chnge in velocity of the cr between t = 1 nd t = 5.

44 Net Chnge Theorem - 4 Problem. If h(t) represents the height of child (in cm) t time t (in yers), nd the child is 120 cm tll t ge 10, which of the following would represent the mount the child grew between t = 10 nd t = 18 yers? A h(t) dt B. C h(t) dt h (t) dt D h (t) dt + 120

45 Net Chnge Theorem - 5 Problem. Suppose wter is flowing into/out of tnk t rte given by r(t) = t L/min, where positive rtes indicte flow in. By how much does the wter level in the tnk chnge during the first 45 minutes fter t = 0?

46 Net Chnge Theorem - 6 Wht is n ssumption you would hve to mke bout the initil mount of wter in the tnk for this to mke sense?

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### Z b. f(x)dx. Yet in the above two cases we know what f(x) is. Sometimes, engineers want to calculate an area by computing I, but... Chpter 7 Numericl Methods 7. Introduction In mny cses the integrl f(x)dx cn be found by finding function F (x) such tht F 0 (x) =f(x), nd using f(x)dx = F (b) F () which is known s the nlyticl (exct) solution.

### ROB EBY Blinn College Mathematics Department ROB EBY Blinn College Mthemtics Deprtment Mthemtics Deprtment 5.1, 5.2 Are, Definite Integrls MATH 2413 Rob Eby-Fll 26 Weknowthtwhengiventhedistncefunction, wecnfindthevelocitytnypointbyfindingthederivtiveorinstntneous

### 5.1 How do we Measure Distance Traveled given Velocity? Student Notes . How do we Mesure Distnce Trveled given Velocity? Student Notes EX ) The tle contins velocities of moving cr in ft/sec for time t in seconds: time (sec) 3 velocity (ft/sec) 3 A) Lel the x-xis & y-xis

### Section 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

### Math 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

### Chapter 8.2: The Integral Chpter 8.: The Integrl You cn think of Clculus s doule-wide 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

### 5.7 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

### Riemann is the Mann! (But Lebesgue may besgue to differ.) Riemnn is the Mnn! (But Lebesgue my besgue to differ.) Leo Livshits My 2, 2008 1 For finite intervls in R We hve seen in clss tht every continuous function f : [, b] R hs the property tht for every ɛ >

### Review 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

### The 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

### Section 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40 Mth B Prof. Audrey Terrs HW # Solutions by Alex Eustis Due Tuesdy, Oct. 9 Section 5. #7,, 6,, 5; Section 5. #8, 9, 5,, 7, 3; Section 5.3 #4, 6, 9, 3, 6, 8, 3; Section 5.4 #7, 8,, 3, 5, 9, 4 5..7 Since

### How 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

### 5.5 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 nti-derivtive is not esily recognizble, then we re in

### a < 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

### Section 6: Area, Volume, and Average Value Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find

### The 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

### Math 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:

### Chapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx... Contents 7.1 Integrtion by Prts................................... 2 7.2 Trigonometric Integrls.................................. 8 7.2.1 Evluting sin m x cos n (x)......................... 8 7.2.2 Evluting

### Riemann 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

### Advanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004 Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when

### Math 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

### P 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0) 1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this

### We divide the interval [a, b] into subintervals of equal length x = b a n Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:

### Math& 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

### Chapter 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

### Definite 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 left-hnd

### Main topics for the Second Midterm Min topics for the Second Midterm The Midterm will cover Sections 5.4-5.9, Sections 6.1-6.3, nd Sections 7.1-7.7 (essentilly ll of the mteril covered in clss from the First Midterm). Be sure to know the

### The Fundamental Theorem of Calculus, Particle Motion, and Average Value The Fundmentl Theorem of Clculus, Prticle Motion, nd Averge Vlue b Three Things to Alwys Keep In Mind: (1) v( dt p( b) p( ), where v( represents the velocity nd p( represents the position. b (2) v ( dt

### 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

### Recitation 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

### x = b a N. (13-1) The set of points used to subdivide the range [a, b] (see Fig. 13.1) is Jnury 28, 2002 13. The Integrl The concept of integrtion, nd the motivtion for developing this concept, were described in the previous chpter. Now we must define the integrl, crefully nd completely. According

### f(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

### Chapter 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

### An Overview of Integration An Overview of Integrtion S. F. Ellermeyer July 26, 2 The Definite Integrl of Function f Over n Intervl, Suppose tht f is continuous function defined on n intervl,. The definite integrl of f from to is

### 7.2 Riemann Integrable Functions 7.2 Riemnn Integrble Functions Theorem 1. If f : [, b] R is step function, then f R[, b]. Theorem 2. If f : [, b] R is continuous on [, b], then f R[, b]. Theorem 3. If f : [, b] R is bounded nd continuous

### W. We shall do so one by one, starting with I 1, and we shall do it greedily, trying Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)

### MAT 168: Calculus II with Analytic Geometry. James V. Lambers MAT 68: Clculus II with Anlytic Geometry Jmes V. Lmbers Februry 7, Contents Integrls 5. Introduction............................ 5.. Differentil Clculus nd Quotient Formuls...... 5.. Integrl Clculus nd

### Objectives. Materials Techer Notes Activity 17 Fundmentl Theorem of Clculus Objectives Explore the connections between n ccumultion function, one defined by definite integrl, nd the integrnd Discover tht the derivtive of the

### Section 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

### AP 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

### Exam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1 Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH-115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixed-point itertion to converge when solving the eqution

### Week 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.

### Section 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

### Final 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, :3-7:pm The following is list of importnt concepts from the sections which were not covered by Midterm Exm or. 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 Clculus nd liner lgebr for biomedicl engineering Week 11: The Riemnn integrl nd its properties Hrtmut Führ fuehr@mth.rwth-chen.de Lehrstuhl A für Mthemtik, RWTH Achen Jnury 9, 2009 Overview 1 Motivtion: