1 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 integrl.
2 Distnce nd Velocity - 1 Distnce nd Velocity Recll tht if we mesure distnce s function of time t, the velocity is determined by differentiting (t), i.e. finding the slope of the grph. Alterntively, suppose we begin with grph of the velocity with respect to time. How cn we determine wht distnce will be trveled? Does it pper in the grph somehow? Let s begin with the simple cse of constnt velocity... distnce = velocity time
3 Distnce nd Velocity - 2 For constnt velocity, the distnce trveled in certin length of time ws simply the re of the rectngle underneth the velocity vs. time grph. Wht if the velocity is chnging? We cn t determine the ect distnce trveled, but mybe we cn estimte it. Let s ssume tht the velocity is not chnging too quickly, so over short mount of time it s roughly constnt. We know how to find the distnce trveled in tht short time... Mking mny of these pproimtions, we could come up with rough estimte of the totl distnce. How does this estimte relte to the grph?
4 Clculting Ares - 1 Clculting Ares It ppers tht the distnce trveled is the re under the grph of velocity, even when the velocity is chnging. We ll see ectly why this is true very soon. If we re simply interested in the re under grph, without ny physicl interprettion, we cn lredy do so if the grph cretes shpe tht we recognize. Emple: Clculte the re between the -is nd the grph of y = + 1 from = 1 nd = 1. y
5 Clculting Ares - 2 Emple: Clculte the re between the -is nd the grph of y = 1 2 from = 1 to = 1. 1 y Wht shpes do you know, right now, for which you cn clculte the ect re?
6 Estimting Ares - 1 Estimting Ares Unfortuntely, mny or most rbitrry res re essentilly impossible to find the re of when the shpe isn t simple composition of tringles, rectngles, or circles. In these cses, we must use less direct methods. We strt by mking n estimte of the re under the grph using shpes whose re is esier to clculte. Suppose we re trying to find the re underneth the grph of the function f() given below between = 1 nd = 4. Drw this re on the grph below, nd lbel it re A
7 Estimting Ares - 2 We cn mke rough estimtion of the re by drwing rectngle tht completely contins the re, or rectngle tht is completely contined by the re. Clculte this overestimte nd underestimte for the re A
8 Estimting Ares - 3 This one-rectngle estimte is very crude. We cn improve our estimtes by using smller rectngles. E.g. We cn divide the intervl from = 1 to = 4 into 3 intervls, ech of width 1, nd use different rectngle heights on ech intervl. Estimte the re A by using 3 rectngles of width 1. Use the function vlue t the left edge of the intervl s the height of ech rectngle
9 Generlizing Are Estimtes - LEFT nd RIGHT - 1 Generlizing Are Estimtes - LEF T (n) nd RIGHT (n) We cn repet the rectngle-building process for ny number of rectngles, nd we epect tht our estimtion of the ect (curved) re will get better the more rectngles we use. The method we used erlier, choosing the height of the rectngles bsed on the function vlue t the left edge, is clled the left hnd sum, nd is denoted LEFT(n) if we use n rectngles. Suppose we re trying to estimte the re under the function f() from = to = b vi the left hnd sum with n rectngles. Then the width of ech rectngle will be = b n. b
10 Generlizing Are Estimtes - LEFT nd RIGHT - 2 If we lbel the endpoints of the intervls to be = 0 < 1 < < n 1 < n = b, then the formul for the left hnd sum will be LEF T (n) =f( 0 ) + f( 1 ) f( n 1 ) n = f( i 1 ). i=1 b n 1 0 n
11 Generlizing Are Estimtes - LEFT nd RIGHT - 3 We hve similr definition for the right hnd sum, or RIGHT(n), clculted by tking the height of ech rectngle to be the height of the function t the right hnd endpoint of the intervl. RIGHT (n) =f( 1 ) + f( 2 ) f( n ) = n f( i ). i=1 b n 1 0 n
12 Generlizing Are Estimtes - LEFT nd RIGHT - 4 Clculte LEFT(6) nd RIGHT(6) for the function shown, between = 1 nd = 4. You will need to estimte some rectngle heights from the grph
13 Generlizing Are Estimtes - LEFT nd RIGHT - 5 In generl, when will LEFT(n) be greter thn RIGHT(n)? When will LEFT(n) be n overestimte for the re? When will LEFT(n) be n underestimte?
14 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( 1) + f( 2) f( n) n = f( i ) where ech i is some point in the intervl [ i 1, i ]. For LEF T (n), we choose the left hnd endpoint of the intervl, so i = i 1 ; for RIGHT (n), we choose the right hnd endpoint, so i = i. i=1 b n 1 0 n
15 Riemnn Sums - 2 The common property of ll these pproimtions is tht they involve sum of rectngulr res, with widths ( ), nd heights (f( i )) There re other Riemnn Sums tht give slightly better estimtes of the re underneth grph, but they often require etr computtion. We will emine some of these other clcultions little lter.
16 The Definite Integrl - 1 The Definite Integrl We observed tht s we increse the number of rectngles used to pproimte the re under curve, our estimte of the re under the grph becomes more ccurte. This implies tht if we wnt to clculte the ect re, we would wnt to use limit. The re underneth the grph of f() between = nd = b is equl to n lim LEF T (n) = lim f( i 1 ), where = b n n n. i=1 b b b b
17 The Definite Integrl - 2 This limit is clled the definite integrl of f() from to b, nd is equl to the re under curve whenever f() is non-negtive continuous function. The definite integrl is written with some specil nottion. Nottion for the Definite Integrl The definite integrl of f() between = nd = b is denoted by the symbol b f() d We cll nd b the limits of integrtion nd f() the integrnd. The d 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 (d), nd heights (f())
18 The Definite Integrl - 3 Emple: Write the definite integrl representing the re underneth the grph of f() = + cos between = 2 nd = 4.
19 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 would give us the sme nswer. Let us look t the left nd right hnd sums for the function 2 on the intervl from = 1 to = 3. Clculte LEF T (2) RIGHT (2) for d. Tht is, how big is the difference between these two estimtes of the re under y = 2 over = ?
20 The Definite Integrl nd LEFT vs RIGHT - 2 Clculte LEF T (4) RIGHT (4) for d.
21 The Definite Integrl nd LEFT vs RIGHT - 3 Clculte LEF T (n) RIGHT (n) for d.
22 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? b f() d = lim n LEF T (n) vs. lim n RIGHT (n)?
23 Estimting Are Between Curves - 1 More on the Definite Integrl nd Are We cn use the definite integrl to clculte other res, s well. Suppose we wnt to find the re between the curves y = 2 nd the line y = 2. It is esy to see tht the two intersect s shown in the following grph. y y = 2 1 y = We cn gin clculte this re by estimting vi rectngles nd the tking the limit to get the definite integrl.
24 Estimting Are Between Curves - 2 y y = 2 1 y = If we estimte this re using the left hnd sum, wht will be the height of the rectngle on the intervl [ i, i+1 ]? 1. height = 2 i (2 i) 2. height = (2 i ) 2 i 3. height = (2 i ) + 2 i
25 Estimting Are Between Curves - 3 Write the formul for LEF T (n), using this height s the function vlue. Wht will be? Write the definite integrl representing the re of this region. 4 3 y 2 y = 2 1 y =
26 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() is lwys negtive? Wht hppens if f() crosses the -is severl times? Emple: 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 net few to B, etc.
27 Negtive Integrl Vlues - 2 Which of these sets of terms hve positive vlues? Which of these sets hve negtive vlues?
28 Negtive Integrl Vlues - 3 Epress the integrl nd E. () b b f(t) dt = A + B + C + D + E f(t) dt in terms of the (positive) res A, B, C, D, (b) (c) (d) b b b f(t) dt = A - B + C - D + E f(t) dt = -A + B - C + D - E f(t) dt = -A - B - C - D - E
29 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.
30 Estimting Integrls - Midpoint Rule - 1 Better Approimtions to Definite Integrls We sw how to pproimte definite integrls using LEF T (n) nd RIGHT (n) Riemnn sums. Unfortuntely, these estimtes re very crude nd inefficient. Even when we hve sophisticted techniques for evluting integrls, these methods will not pply to ll functions, for emple: e 2 d - used in probbility sin() d - used in optics
31 Estimting Integrls - Midpoint Rule - 2 To evlute definite integrls of such functions, we could use left or right hnd Riemnn sums. However, it would be preferble to develop more ccurte estimtes. But more ccurte estimtes cn lwys be mde by using more rectngles. More precisely, we wnt to develop more efficient estimtes: estimtes tht re more ccurte for similr mounts of work.
32 Estimting Integrls - Midpoint Rule - 3 Midpoint Rule The ccurcy of Riemnn sum clcultion will usully improve if we choose the midpoint of ech subdivision rther thn the right or left endpoints. y y y b b b Compre the ccurcy of the left-hnd, right-hnd nd midpoint rules for estimting the re on the intervl.
33 Estimting Integrls - Midpoint Rule - 4 For wht kinds of functions f will the midpoint rule lwys give vlue tht is ectly equl to the integrl?
34 Estimting Integrls - Trpezoidl Rule - 1 The Trpezoidl Rule The Midpoint Rule is only one possible vrition on Riemnn sums. Another pproch is to use shpe other thn rectngle to estimte the re on n intervl. For the ppropritely nmed Trpezoidl Rule, we use trpezoid on ech intervl. Sketch trpezoidl pproimtion to the re under the grph, nd then write formul for the re of the single trpezoid. y i 1 i
35 Estimting Integrls - Trpezoidl Rule - 2 Write formul for the full Trpezoid Rule (written T RAP (n)), estimting the entire re under grph with multiple intervls. b n 1 0 n
36 Estimting Integrls - Trpezoidl Rule - 3 Alterntive: Epress the trpezoidl rule T RAP (n) in terms of the left nd right hnd Riemnn sums (LEF T (n) nd RIGHT (n)). b n 1 0 n
37 Estimting Integrls - Trpezoidl Rule - 4 The trpezoidl rule is especilly ccurte for functions f for which f () is smll for ll. Eplin from n intuitive point of view why you would epect this sttement to be correct. b n 1 n
38 Estimting Integrls - Trpezoidl Rule Emples - 1 Use the trpezoidl rule to estimte vlues in the following tble for f() f() f() d, if we hve mesured the f()
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
Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn
Mth 4 Lb 1 Prt II: Numericl Integrtion This section includes severl techniques for getting pproimte numericl vlues for definite integrls without using ntiderivtives. Mthemticll, ect nswers re preferble
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
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
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
Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished
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
Mth 43 Section 6. Section 6.: Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot
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
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
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
_.qd // : PM Pge 9 SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion
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
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
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
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
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
4 pplictions of definite integrls 5. Volumes: Disks nd Wshers In the previous section, we computed volumes of solids for which we could determine the re of cross-section or slice. In this section, we restrict
Mth 43 Section 4839 M TH 4: PM 6: PM Susn Wheeler firstname.lastname@example.org 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
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
Curve on Clcultor eperience Fin n ownlo (or type in) progrm on your clcultor tht will fin the re uner curve using given number of rectngles. Mke sure tht the progrm fins LRAM, RRAM, n MRAM. (You nee to
The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Tody we provide the connection
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
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,
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:
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
Curve on 6 For -, () Evlute the integrl, n (b) check your nswer by ifferentiting. ( ). ( ). ( ).. 6. sin cos 7. sec csccot 8. sec (sec tn ) 9. sin csc. Evlute the integrl sin by multiplying the numertor
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 >
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
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
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
Nme: Algebr II Honors Pre-Chpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble
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
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 non-liner, we hd to work hrd limits derivtive A relted question is the re under f(x) (but
.6 Numericl Integrtion 5.6 Numericl Integrtion Approimte definite integrl using the Trpezoidl Rule. Approimte definite integrl using Simpson s Rule. Anlze the pproimte errors in the Trpezoidl Rule nd Simpson
Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 2-3, Sections 4.1-4.8, nd Sections 5.1-5.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the
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 x-xis etween nd is denoted y f(x) dx nd clled the
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
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
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
. 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
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
MA 124 Jnury 18, 2018 Prof PB s one-minute introduction to clculus Derivtives re. Integrls re. In Clculus 1, we lern limits, derivtives, some pplictions of derivtives, indefinite integrls, definite integrls,
5.1 ESTIMATING WITH FINITE SUMS Emple: Suppose from the nd to 4 th hour of our rod trip, ou trvel with the cruise control set to ectl 70 miles per hour for tht two hour stretch. How fr hve ou trveled during
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.
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.
4.5 The Funmentl Theorem of Clculus Contemporry Clculus 4.5 THE FUNDAMENTAL THEOREM OF CALCULUS This section contins the most importnt n most use theorem of clculus, THE Funmentl Theorem of Clculus. Discovere
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
A-Level Mthemtics Trnsition Tsk (compulsory for ll mths students nd ll further mths student) Due: st Lesson of the yer. Length: - hours work (depending on prior knowledge) This trnsition tsk provides revision
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
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
38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the x-xis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These
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
5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce
4. Are under Curve A cr is trveling so tht its speed is never decresing during 1-second intervl. The speed t vrious moments in time is listed in the tle elow. Time in Seconds 3 6 9 1 Speed in t/sec 3 37
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
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
Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn
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
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
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
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
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
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
Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 6.5 Numericl Approximtions of Definite Integrls Sometimes the integrl of function cnnot be expressed with elementry functions, i.e., polynomil,
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
Summer 6 MTH4 College Clculus Section J Lecture Notes Yin Su University t Bufflo email@example.com Contents Bsic techniques of integrtion 3. Antiderivtive nd indefinite integrls..............................................
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
To compute n re we pproimte region b rectngles nd let the number of rectngles become lrge. The precise re is the limit of these sums of res of rectngles. Integrls Now is good time to red (or rered) A Preview
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
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
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
7 Integrtion º½ ÌÛÓ Ü ÑÔÐ Up to now we hve been concerned with extrcting informtion bout how function chnges from the function itself. Given knowledge bout n object s position, for exmple, we wnt to know
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)
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 t-is, n between t n t. (You my wnt to look up the re formul