(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P.

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P."

Transcription

1 Chpter 7: The Riemnn Integrl When the derivtive is introdued, it is not hrd to see tht the it of the differene quotient should be equl to the slope of the tngent line, or when the horizontl xis is time nd the vertil is distne, equl to the instntneous veloity. But the integrl is not so esily interpreted: Why is the re under the urve in ny wy relted to the ntiderivtive? It relly ws stroke of genius by Newton nd Leibnitz to see tht onnetion. In ft, integrtion, in the sense of re or volume, ws invented first, mny enturies erlier: Arhimedes ws involved in trying to ompute the volume of wine sk. The derivtive ws muh hrder onept. In ft, Zeno s prdoxes show how hrd the Greeks found the onept of it in generl. Differentition ould relly only be reognized nd investigted fter Desrtes invented oordinte geometry. As result, few lulus texts, seeking historil pproh to the subjet, over integrtion before differentition. Definition. Let f be bounded funtion on ompt intervl [, b]. (We ll see lter why the underlined words re importnt. For now, just note tht we re not ssuming tht f is even ontinuous. ( A prtition P of [, b] is finite subset of [, b] ontining nd b. If Q is nother prtition nd P Q, then Q is refinement of P. (b Write P = {x 0, x, x 2,..., x n } where = x 0 < x < x 2 < < x n = b. Then U(f, P = L(f, P = n sup{f(x : x i x x i }(x i x i i= n inf{f(x : x i x x i }(x i x i i= re the upper nd lower sums with respet to f nd P. ( The upper nd lower integrls of f on [, b] re U(f = inf{u(f, P } nd L(f = sup{l(f, P }, where the inf nd sup re tken over ll prtitions P of [, b]. (d If U(f = L(f, then f is integrble on [, b], nd their ommon vlue is denoted b f or f(x dx. b Exmple. On the intervl [, b] with prtition P s shown, onsider the funtion f. We hve U(f, P = N(x x 0 + M(x 2 x + N(x 3 x 2 + M(x 4 x 3 + M(x 5 x 4 L(f, P = 0(x x 0 + N(x 2 x + 0(x 3 x 2 + N(x 4 x 3 + 0(x 5 x 4 The blue region shows the re representing the upper sum, nd the pink region the lower sum.

2 With prtition Q tht refines P, the upper sum dereses nd the lower sum inreses: Repling the prtitions with ever finer ones, we see tht the upper sums nd lower sums pproh eh other, with the re of the three tringles s the ommon it: U(f = L(f = b f.

3 We see now why we restrited to bounded funtions: So tht the sup nd inf of f on eh subintervl exists (nd is finite. And similrly we restrited to bounded intervls so tht we do not hve infinitely long subintervls or infinitely mny subintervls, mking the upper nd lower sums muh hrder or impossible to interpret. (We restrited to losed intervls so tht the funtion hs vlues t the endpoints. Lter in lulus ourse integrls of unbounded funtions or over unbounded or unlosed intervls might be llowed, but suh things re lwys lled improper integrls nd interpreted s its of proper integrls, its tht my or my not exist: Exmple. Improper integrls: 0 dx = x ε 0 + x dx = M sin x dx =? M ε M x dx = M The orret version of the lst one is M x dx = sin x dx = [ 2x /2] ε 0 + M [2x /2] M = 2( ε = 2 ε ε 0 + = M 2( M = [ os M x]m M = 0 = 0 M sin x dx = N M M N sin x dx = N [ os M x]m N does not exist The it where the left nd right endpoint pproh nd t the sme rte is the lled the prinipl vlue of the improper integrl. Exmple. A funtion tht is not integrble: The Dirihlet funtion χ Q on [0, ]. Every subintervl in every prtition ontins rtionl numbers, so the supremum of the χ Q -vlues on the subintervl is, so the upper sum for every prtition is, so the upper integrl is. But every subintervl in every prtition lso ontins irrtionl numbers, so the infimum of the χ Q -vlues on the subintervl is 0, so the lower sum for every prtition is 0, so the lower integrl is 0. Lemm. For ny bounded funtion f on the ompt intervl [, b]: ( For ny prtition P of [, b], L(f, P U(f, P. (b For ny prtitions P, Q of [, b], if P Q, then L(f, P L(f, Q nd U(f, P U(f, Q. ( For ny prtition P of [, b], L(f, P U(f Therefore, L(f U(f. Proof. ( On ny subintervl [x i, x i ], inf{f(x : x i x x i } sup{f(x : x i x x i }. (b It is enough to ssume tht Q hs only one more element thn P, sy P = {x 0, x,..., x n } nd Q hs the dditionl division point x between x j nd x j. Then p = inf{f(x : x j x

4 x j } is less thn or equl to eh of q = inf{f(x : x j x x } nd q 2 = inf{f(x : x x x j } it is equl to t lest one, but no lrger thn either. So the term p(x j x j in L(f, P tht orresponds to [x j, x j ] is t most the sum q (x x j + q 2 (x j x of the two terms in L(f, Q tht orrespond to [x j, x ] nd [x, x j ] beuse x j x j = (x x j + (x j x. Adding up ll the terms for ll the subintervls in eh prtition, we see tht L(f, P L(f, Q. Similrly, U(f, P U(f, Q. ( Assume BWOC tht there is prtition P for whih L(f, P > U(f. Then L(f, P is not lower bound for the set of ll upper sums, so there is prtition Q for whih L(f, P > U(f, Q. But then P Q is refinement of both P nd Q, so by (b nd (, we hve L(f, P L(f, P Q U(f, P Q U(f, Q, ontrdition. Thus, U(f is n upper bound on the set of ll lower sums, it is t lest s lrge s the lest upper bound, nd we hve the Therefore sentene. Here is nother wy to sy integrble, in whih, insted of finding one prtition P tht mkes L(f, P lmost s lrge s possible nd nother, Q, tht mkes U(f, P lmost s smll s possible, we sy tht it is enough to find single prtition tht mkes the upper nd lower sums lose to eh other: Proposition. A bounded funtion f : [, b] R is integrble iff, ε > 0, P prtition of [, b] s.t. U(f, P L(f, P < ε. Proof. ( Assume BWOC tht U(f L(f. Then then U(f L(f = ε > 0. By hypothesis, P s.t. U(f, P L(f, P < ε. But beuse L(f, P L(f nd U(f, P U(f, we hve U(f L(f U(f, P L(f, P < ε = U(f L(f, /\. ( Let ε > 0 be given, nd pik P, Q s.t. U(f, P b f < ε/2 nd b f L(f, Q < ε/2 Then beuse P Q is ommon refinement of P, Q, we hve so L(f, Q L(f, P Q U(f, P Q U(f, P, U(f, P Q L(f, P Q U(f, P L(f, Q ( b = U(f, P f + ( b f L(f, Q < ε. Now we show tht ontinuous funtion (on ompt intervl is integrble. Beuse not ll ontinuous funtions re differentible, we see tht it is hrder for funtion to be differentible thn for it to be integrble. Theorem. If f : [, b] R is ontinuous, then it is integrble. Proof. Let ε > 0 be given. Beuse f is uniformly ontinuous, there is δ > 0 for whih x i x j < δ implies f(x i f(x j < ε/(b. Pik prtition P for whih ll the subintervls [x i, x i ] hve length less thn δ. (For exmple, we might hoose n in N so lrge tht (b /n < δ nd set P = {x i = +i(b /n : i = 0,,..., n}. Then beuse f ttins its inf nd sup on eh [x i, x i ], sy t t i nd u i respetively, we hve t i u i x i x i < δ, so f(t i f(u i < ε/(b, so U(f, P L(f, P = n (f(t i f(u i (x i x i < i= n i= ε b (x i x i = ε b (x n x 0 = ε,

5 the seond lst equlity beuse the sum (x x 0 + (x 2 x + + (x n x n telesopes, i.e., ll the terms nel exept the seond nd seond lst. But there re mny disontinuous integrble funtions; our first exmple (the three tringles ws disontinuous t two points but still integrble. We hve seen tht the Dirihlet funtion on [0, ] is not integrble. But we tht the Thome funtion is integrble on this intervl: Exmple. Rell tht the Thome funtion is given by { 0 if x is irrtionl t(x = n if x = m n in lowest terms Beuse every subintervl of every prtition of [0, ] ontins irrtionl numbers, the lower sum of t with respet to every prtition is 0, so the lower integrl of t is 0. Thus, to see tht t is integrble, we need to show tht the upper integrl is 0, i.e., tht, given ε > 0, we n find prtition P with respet to whih U(t, P < ε. We n do tht: There re only finitely mny rtionls x in [0, ] for whih f(x > ε/2. Pik prtition P of [0, ] so tht these finitely mny rtionls re the enters (or, for x = 0 nd x =, the ends of subintervls with totl length ε/2. Then in U(f, P, the terms orresponding to those subintervls dd up to t most times the totl length of these subintervls nd hene less thn ε/2. In the other subintervls the vlue of the funtion is t most ε/2, so those subintervls ontribute to U(f, P totl of less thn (ε/2. It follows tht U(f, P < ε. In ft, it is shown in the text tht bounded funtion on ompt intervl is integrble iff its set of disontinuities hs mesure zero. We won t try to explin this term, beuse the right wy to do tht is to tke totlly different pproh to the ourse, in terms of Lebesgue integrtion. So we will only sy tht ll ountble sets, inluding finite sets, nd the Cntor set hve mesure zero, so tht funtions tht re disontinuous only on these sets re integrble. ( Mesure is onept tht extends the ide of length, so set of length l hs mesure l; but it is hrd to ssign length to set like the rtionls, whih hs mesure 0. Remember tht the Dirihlet funtion is disontinuous everywhere, so its set of disontinuities in [0, ] hs mesure one; but the Thome funtion is disontinuous only on the rtionls, whih hve mesure 0. And sure enough, the Dirihlet funtion is not integrble, while the Thome funtion is. [At this point students re redy to do the eleventh problem set.] Proposition. (Properties of the definite integrl Suppose f, g re integrble on [, b] nd k is onstnt. Then: (0 If < < b, then b f = f + b f. ( b (f + g = b f + b g. (2 b kf = k b f. (3 If f g on [, b], then b f b g. Corollry. (of (3: (3 If m f M on [, b], then m(b b f M(b. (3b f is integrble nd b f b f.

6 The ft tht f is integrble is new informtion the proof is n exerise. To do it, you should verify tht, if m i, M i re the inf nd sup of f on [x i, x i ] nd m i, M i re the sme for f, then M i m i M i m i. To get the seond ssertion in (3b, use f f f on [.b]. Remrks on the proposition: (0 Prt of wht is to be proved here is tht f is integrble on [, b] iff it is integrble on both [, ] nd [, b]. But if we define f = 0 nd b f = b f, then we don t need to dd the If prt of this sttement s long s f is integrble on the intervls determined by, b,. ( The proof is n exerise. (2 Here is proof of the k < 0 se: Let ε > 0 be given, nd pik prtition P of [, b] for whih U(f, P b f < ε/(2 k nd b f L(f, P < ε/(2 k. Then beuse, on eh subintervl [x i, x i ] in P we hve sup{kf(x : x i x x i } = k inf{f(x : x i x x i } nd similrly with sup nd inf reversed, it follows tht U(kf, P L(kf, P = kl(f, P ku(f, P = k (U(f, P L(f, P ( b b k U(f, P f + f L(f, P < ε 2 + ε 2 = ε, so kf is integrble, nd U(kf, P k so k b f = b kf.// b ( f = k L(f, P b ( b f = k f L(f, P < ε 2 < ε, (3 Beuse f g, for ny subintervl [x i, x i ] of ny prtition P of [, b] we hve sup{f(x : x i x x i } sup{g(x : x i x x i } so U(f, P U(g, P, so b f = inf P U(f, P inf P U(g, P = b g. [At this point students re redy to do the twelfth problem set.] Theorem. (Fundmentl Theorem of Clulus ( If f is the derivtive of F on [, b] nd f is integrble, then b f = F (b F (. (2 Let g : [, b] R be integrble, nd define G(t = t g(x dx. Then G : [, b] R is ontinuous. If g is ontinuous t in [, b], then G is differentible t nd G ( = g(. Proof. ( For ny subintervl [x i, x i ] in ny prtition P of [, b], by the Men Vlue Theorem x i [x i, x i ] s.t. F (x i F (x i = f(x i (x i x i, so n i= (F (x i F (x i is between L(f, P nd U(f, P. But the sum is telesoping, to F (b F (, independent of the prtition P, so F (b F ( is between L(f nd U(f, whih re both equl to b f.

7 (2 In ft, we n show tht G is Lipshitz, whih is stronger thn ontinuous: Let M be n upper bound on g in [, b]. Then for ll x, x 2 [, b], we hve 2 x G(x G(x 2 = g g = g M x x 2. Now suppose g is ontinuous t. Then G G(x G( ( = = g x x x x, so we wnt to show, given ε > 0, δ > 0 s.t. 0 < x < δ = g/(x g( < ε. Now there is δ > 0 s.t., if x < δ, then g(x g( < ε, nd for tht δ, if 0 < x < δ, then g ( x g( = x (g(t g(dt x g(t g( dt ε dt = ε. x x (In the middle of this, note tht, if x < 0, then ny x 2 of nonnegtive funtion is lso 0. Here re some exmples of funtions g nd the funtions G defined by their integrls, G(x = g(t dt. Exmple. In these two exmples, = 2, so tht, no mtter wht g is, G( 2 = 2 2 g = 0. We re interested espeilly in the points t whih the definitions of g hnge from one formul to nother. First: { if x < 0 g(x =, if x 0 { G(x = g(t dt = 2 dt = x 2 if x < 0 2 G(0 + 0 dt = 2 + x if x 0 Here, G is differentible exept t the disontinuity x = 0 of g. Seond: { if x g(x =, 2x 3 if x > { x 2 if x G(x = g(t dt = 2 G( + (2t 3dt = 3 + (x2 3x ( 2 = x 2 3x if x > Beuse this g is ontinuous t the point where its definition hnges, the orresponding G is differentible there. Here re the orresponding grphs, with the g s in green nd the G s in red:

8 Though our text doesn t seem to inlude them, I hve seen in other texts exmples of funtions defined by integrls where the upper, nd mybe lso the lower, it(s of integrtion re themselves funtions. In order to find their derivtives, we only need to throw in the Chin Rule: Suppose we re given H(x = f(x g(t dt. We n isolte n intervening funtion: G(u = u g(t dt. Then H(x = G(f(x, so (d/dx(h(x = G (f(x f (x = g(f(xf (x. Exmple. d dx 5x x ( exp( t 2 dt = d 5x 2 +3 exp( t 2 dt dx 0 2x 0 exp( t 2 dt = exp( (5x (0x exp( (2x 2 2 = 0x exp( (5x exp( (2x 2. Here, the intervening funtion is G(u = u 0 exp( t2 dt, so tht G (u = exp( u 2. There is nothing speil bout 0; beuse exp( t 2 is ontinuous everywhere, ny onstnt would hve worked. [At this point students re redy to do the thirteenth problem set.]

7.2 Riemann Integrable Functions

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

More information

Part 4. Integration (with Proofs)

Part 4. Integration (with Proofs) Prt 4. Integrtion (with Proofs) 4.1 Definition Definition A prtition P of [, b] is finite set of points {x 0, x 1,..., x n } with = x 0 < x 1

More information

T b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions.

T b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions. Rel Vribles, Fll 2014 Problem set 5 Solution suggestions Exerise 1. Let f be bsolutely ontinuous on [, b] Show tht nd T b (f) P b (f) f (x) dx [f ] +. Conlude tht if f is in AC then it is the differene

More information

The Regulated and Riemann Integrals

The 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 information

The Riemann-Stieltjes Integral

The Riemann-Stieltjes Integral Chpter 6 The Riemnn-Stieltjes Integrl 6.1. Definition nd Eistene of the Integrl Definition 6.1. Let, b R nd < b. ( A prtition P of intervl [, b] is finite set of points P = { 0, 1,..., n } suh tht = 0

More information

Chapter 6. Riemann Integral

Chapter 6. Riemann Integral Introduction to Riemnn integrl Chpter 6. Riemnn Integrl Won-Kwng Prk Deprtment of Mthemtics, The College of Nturl Sciences Kookmin University Second semester, 2015 1 / 41 Introduction to Riemnn integrl

More information

Math 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1)

Math 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1) Green s Theorem Mth 3B isussion Session Week 8 Notes Februry 8 nd Mrh, 7 Very shortly fter you lerned how to integrte single-vrible funtions, you lerned the Fundmentl Theorem of lulus the wy most integrtion

More information

MA10207B: ANALYSIS SECOND SEMESTER OUTLINE NOTES

MA10207B: ANALYSIS SECOND SEMESTER OUTLINE NOTES MA10207B: ANALYSIS SECOND SEMESTER OUTLINE NOTES CHARLIE COLLIER UNIVERSITY OF BATH These notes hve been typeset by Chrlie Collier nd re bsed on the leture notes by Adrin Hill nd Thoms Cottrell. These

More information

Review of Riemann Integral

Review 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 information

More Properties of the Riemann Integral

More Properties of the Riemann Integral More Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologil Sienes nd Deprtment of Mthemtil Sienes Clemson University Februry 15, 2018 Outline More Riemnn Integrl Properties The Fundmentl

More information

MAT 403 NOTES 4. f + f =

MAT 403 NOTES 4. f + f = MAT 403 NOTES 4 1. Fundmentl Theorem o Clulus We will proo more generl version o the FTC thn the textook. But just like the textook, we strt with the ollowing proposition. Let R[, ] e the set o Riemnn

More information

INTEGRATION. 1 Integrals of Complex Valued functions of a REAL variable

INTEGRATION. 1 Integrals of Complex Valued functions of a REAL variable INTEGRATION NOTE: These notes re supposed to supplement Chpter 4 of the online textbook. 1 Integrls of Complex Vlued funtions of REAL vrible If I is n intervl in R (for exmple I = [, b] or I = (, b)) nd

More information

Chapter 0. What is the Lebesgue integral about?

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

More information

Presentation Problems 5

Presentation Problems 5 Presenttion Problems 5 21-355 A For these problems, ssume ll sets re subsets of R unless otherwise specified. 1. Let P nd Q be prtitions of [, b] such tht P Q. Then U(f, P ) U(f, Q) nd L(f, P ) L(f, Q).

More information

RIEMANN INTEGRATION. Throughout our discussion of Riemann integration. B = B [a; b] = B ([a; b] ; R)

RIEMANN INTEGRATION. Throughout our discussion of Riemann integration. B = B [a; b] = B ([a; b] ; R) RIEMANN INTEGRATION Throughout our disussion of Riemnn integrtion B = B [; b] = B ([; b] ; R) is the set of ll bounded rel-vlued funtons on lose, bounded, nondegenerte intervl [; b] : 1. DEF. A nite set

More information

For a continuous function f : [a; b]! R we wish to define the Riemann integral

For a continuous function f : [a; b]! R we wish to define the Riemann integral Supplementry Notes for MM509 Topology II 2. The Riemnn Integrl Andrew Swnn For continuous function f : [; b]! R we wish to define the Riemnn integrl R b f (x) dx nd estblish some of its properties. This

More information

The Riemann and the Generalised Riemann Integral

The Riemann and the Generalised Riemann Integral The Riemnn nd the Generlised Riemnn Integrl Clvin 17 July 14 Contents 1 The Riemnn Integrl 1.1 Riemnn Integrl............................................ 1. Properties o Riemnn Integrble Funtions.............................

More information

6.5 Improper integrals

6.5 Improper integrals Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =

More information

University of Sioux Falls. MAT204/205 Calculus I/II

University of Sioux Falls. MAT204/205 Calculus I/II University of Sioux Flls MAT204/205 Clulus I/II Conepts ddressed: Clulus Textook: Thoms Clulus, 11 th ed., Weir, Hss, Giordno 1. Use stndrd differentition nd integrtion tehniques. Differentition tehniques

More information

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the

More information

n 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

n 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 information

f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as f(x) dx = lim f(x i ) x; i=1

f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as f(x) dx = lim f(x i ) x; 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 information

Lecture 1: Introduction to integration theory and bounded variation

Lecture 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 information

MATH 409 Advanced Calculus I Lecture 19: Riemann sums. Properties of integrals.

MATH 409 Advanced Calculus I Lecture 19: Riemann sums. Properties of integrals. MATH 409 Advnced Clculus I Lecture 19: Riemnn sums. Properties of integrls. Drboux sums Let P = {x 0,x 1,...,x n } be prtition of n intervl [,b], where x 0 = < x 1 < < x n = b. Let f : [,b] R be bounded

More information

The Henstock-Kurzweil integral

The Henstock-Kurzweil integral fculteit Wiskunde en Ntuurwetenschppen The Henstock-Kurzweil integrl Bchelorthesis Mthemtics June 2014 Student: E. vn Dijk First supervisor: Dr. A.E. Sterk Second supervisor: Prof. dr. A. vn der Schft

More information

Main topics for the First Midterm

Main topics for the First Midterm 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

More information

MATH Final Review

MATH Final Review MATH 1591 - Finl Review November 20, 2005 1 Evlution of Limits 1. the ε δ definition of limit. 2. properties of limits. 3. how to use the diret substitution to find limit. 4. how to use the dividing out

More information

Math 554 Integration

Math 554 Integration Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we

More information

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

Properties 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 information

Riemann is the Mann! (But Lebesgue may besgue to differ.)

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 >

More information

MATH 409 Advanced Calculus I Lecture 22: Improper Riemann integrals.

MATH 409 Advanced Calculus I Lecture 22: Improper Riemann integrals. MATH 409 Advned Clulus I Leture 22: Improper Riemnn integrls. Improper Riemnn integrl If funtion f : [,b] R is integrble on [,b], then the funtion F(x) = x f(t)dt is well defined nd ontinuous on [,b].

More information

11 An introduction to Riemann Integration

11 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 information

4.4 Areas, Integrals and Antiderivatives

4.4 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 information

Riemann Sums and Riemann Integrals

Riemann 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 information

LECTURE. INTEGRATION AND ANTIDERIVATIVE.

LECTURE. INTEGRATION AND ANTIDERIVATIVE. ANALYSIS FOR HIGH SCHOOL TEACHERS LECTURE. INTEGRATION AND ANTIDERIVATIVE. ROTHSCHILD CAESARIA COURSE, 2015/6 1. Integrtion Historiclly, it ws the problem of computing res nd volumes, tht triggered development

More information

Advanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004

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

More information

IMPORTANT THEOREMS CHEAT SHEET

IMPORTANT THEOREMS CHEAT SHEET IMPORTANT THEOREMS CHEAT SHEET BY DOUGLAS DANE Howdy, I m Bronson s dog Dougls. Bronson is still complining bout the textbook so I thought if I kept list of the importnt results for you, he might stop.

More information

7.2 The Definition of the Riemann Integral. Outline

7.2 The Definition of the Riemann Integral. Outline 7.2 The Definition of the Riemnn Integrl Tom Lewis Fll Semester 2014 Upper nd lower sums Some importnt theorems Upper nd lower integrls The integrl Two importnt theorems on integrbility Outline Upper nd

More information

Principles of Real Analysis I Fall VI. Riemann Integration

Principles of Real Analysis I Fall VI. Riemann Integration 21-355 Principles of Rel Anlysis I Fll 2004 A. Definitions VI. Riemnn Integrtion Let, b R with < b be given. By prtition of [, b] we men finite set P [, b] with, b P. The set of ll prtitions of [, b] will

More information

Riemann Sums and Riemann Integrals

Riemann 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 information

Lecture 1. Functional series. Pointwise and uniform convergence.

Lecture 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 information

INTRODUCTION TO INTEGRATION

INTRODUCTION 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 information

AP Calculus AB Unit 4 Assessment

AP Calculus AB Unit 4 Assessment Clss: Dte: 0-04 AP Clulus AB Unit 4 Assessment Multiple Choie Identify the hoie tht best ompletes the sttement or nswers the question. A lultor my NOT be used on this prt of the exm. (6 minutes). The slope

More information

1 The Riemann Integral

1 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 information

arxiv: v1 [math.ca] 21 Aug 2018

arxiv: v1 [math.ca] 21 Aug 2018 rxiv:1808.07159v1 [mth.ca] 1 Aug 018 Clulus on Dul Rel Numbers Keqin Liu Deprtment of Mthemtis The University of British Columbi Vnouver, BC Cnd, V6T 1Z Augest, 018 Abstrt We present the bsi theory of

More information

MATH 409 Advanced Calculus I Lecture 18: Darboux sums. The Riemann integral.

MATH 409 Advanced Calculus I Lecture 18: Darboux sums. The Riemann integral. MATH 409 Advnced Clculus I Lecture 18: Drboux sums. The Riemnn integrl. Prtitions of n intervl Definition. A prtition of closed bounded intervl [, b] is finite subset P [,b] tht includes the endpoints

More information

MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

MA123, 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. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

More information

Unit #9 : Definite Integral Properties; Fundamental Theorem of Calculus

Unit #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 information

Lecture 3 ( ) (translated and slightly adapted from lecture notes by Martin Klazar)

Lecture 3 ( ) (translated and slightly adapted from lecture notes by Martin Klazar) Lecture 3 (5.3.2018) (trnslted nd slightly dpted from lecture notes by Mrtin Klzr) Riemnn integrl Now we define precisely the concept of the re, in prticulr, the re of figure U(, b, f) under the grph of

More information

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019

ACCESS 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 information

Properties of the Riemann Integral

Properties of the Riemann Integral Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University Februry 15, 2018 Outline 1 Some Infimum nd Supremum Properties 2

More information

Week 10: Riemann integral and its properties

Week 10: Riemann integral and its properties Clculus nd Liner Algebr for Biomedicl Engineering Week 10: Riemnn integrl nd its properties H. Führ, Lehrstuhl A für Mthemtik, RWTH Achen, WS 07 Motivtion: Computing flow from flow rtes 1 We observe the

More information

MAA 4212 Improper Integrals

MAA 4212 Improper Integrals Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly well-defined, is too restrictive for mny purposes; there re functions which

More information

Stuff You Need to Know From Calculus

Stuff 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 information

MA 124 January 18, Derivatives are. Integrals are.

MA 124 January 18, Derivatives are. Integrals are. 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,

More information

Review of Calculus, cont d

Review of Calculus, cont d Jim Lmbers MAT 460 Fll Semester 2009-10 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 information

Calculus I-II Review Sheet

Calculus I-II Review Sheet Clculus I-II Review Sheet 1 Definitions 1.1 Functions A function is f is incresing on n intervl if x y implies f(x) f(y), nd decresing if x y implies f(x) f(y). It is clled monotonic if it is either incresing

More information

Calculus in R. Chapter Di erentiation

Calculus in R. Chapter Di erentiation Chpter 3 Clculus in R 3.1 Di erentition Definition 3.1. Suppose U R is open. A function f : U! R is di erentible t x 2 U if there exists number m such tht lim y!0 pple f(x + y) f(x) my y =0. If f is di

More information

The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O

The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O 1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the

More information

Overview of Calculus I

Overview 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 information

W. We shall do so one by one, starting with I 1, and we shall do it greedily, trying

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)

More information

A 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. 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 information

CHAPTER V INTEGRATION, AVERAGE BEHAVIOR A = πr 2.

CHAPTER V INTEGRATION, AVERAGE BEHAVIOR A = πr 2. CHAPTER V INTEGRATION, AVERAGE BEHAVIOR A πr 2. In this hpter we will derive the formul A πr 2 for the re of irle of rdius r. As mtter of ft, we will first hve to settle on extly wht is the definition

More information

FUNDAMENTALS OF REAL ANALYSIS by. III.1. Measurable functions. f 1 (

FUNDAMENTALS OF REAL ANALYSIS by. III.1. Measurable functions. f 1 ( FUNDAMNTALS OF RAL ANALYSIS by Doğn Çömez III. MASURABL FUNCTIONS AND LBSGU INTGRAL III.. Mesurble functions Hving the Lebesgue mesure define, in this chpter, we will identify the collection of functions

More information

MAT612-REAL ANALYSIS RIEMANN STIELTJES INTEGRAL

MAT612-REAL ANALYSIS RIEMANN STIELTJES INTEGRAL MAT612-REAL ANALYSIS RIEMANN STIELTJES INTEGRAL DR. RITU AGARWAL MALVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR, INDIA-302017 Tble of Contents Contents Tble of Contents 1 1. Introduction 1 2. Prtition

More information

Math Calculus with Analytic Geometry II

Math 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 x-xis) ( bove f under x-xis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem

More information

38 Riemann sums and existence of the definite integral.

38 Riemann sums and existence of the definite integral. 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

More information

UNIFORM 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 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 information

Appendix to Notes 8 (a)

Appendix to Notes 8 (a) Appendix to Notes 8 () 13 Comprison of the Riemnn nd Lebesgue integrls. Recll Let f : [, b] R be bounded. Let D be prtition of [, b] such tht Let D = { = x 0 < x 1

More information

Line Integrals and Entire Functions

Line Integrals and Entire Functions Line Integrls nd Entire Funtions Defining n Integrl for omplex Vlued Funtions In the following setions, our min gol is to show tht every entire funtion n be represented s n everywhere onvergent power series

More information

UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE

UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE 1. Pointwise Convergence of Sequence Let E be set nd Y be metric spce. Consider functions f n : E Y for n = 1, 2,.... We sy tht the sequence

More information

Calculus Cheat Sheet. Integrals Definitions. where F( x ) is an anti-derivative of f ( x ). Fundamental Theorem of Calculus. dx = f x dx g x dx

Calculus Cheat Sheet. Integrals Definitions. where F( x ) is an anti-derivative of f ( x ). Fundamental Theorem of Calculus. dx = f x dx g x dx Clulus Chet Sheet Integrls Definitions Definite Integrl: Suppose f ( ) is ontinuous Anti-Derivtive : An nti-derivtive of f ( ) on [, ]. Divide [, ] into n suintervls of is funtion, F( ), suh tht F = f.

More information

Type 2: Improper Integrals with Infinite Discontinuities

Type 2: Improper Integrals with Infinite Discontinuities mth imroer integrls: tye 6 Tye : Imroer Integrls with Infinite Disontinuities A seond wy tht funtion n fil to be integrble in the ordinry sense is tht it my hve n infinite disontinuity (vertil symtote)

More information

Formula for Trapezoid estimate using Left and Right estimates: Trap( n) If the graph of f is decreasing on [a, b], then f ( x ) dx

Formula for Trapezoid estimate using Left and Right estimates: Trap( n) If the graph of f is decreasing on [a, b], then f ( x ) dx Fill in the Blnks for the Big Topis in Chpter 5: The Definite Integrl Estimting n integrl using Riemnn sum:. The Left rule uses the left endpoint of eh suintervl.. The Right rule uses the right endpoint

More information

Math 8 Winter 2015 Applications of Integration

Math 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 information

6.1 Definition of the Riemann Integral

6.1 Definition of the Riemann Integral 6 The Riemnn Integrl 6. Deinition o the Riemnn Integrl Deinition 6.. Given n intervl [, b] with < b, prtition P o [, b] is inite set o points {x, x,..., x n } [, b], lled grid points, suh tht x =, x n

More information

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60.

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60. Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23

More information

f(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

f(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 information

The Double Integral. The Riemann sum of a function f (x; y) over this partition of [a; b] [c; d] is. f (r j ; t k ) x j y k

The Double Integral. The Riemann sum of a function f (x; y) over this partition of [a; b] [c; d] is. f (r j ; t k ) x j y k The Double Integrl De nition of the Integrl Iterted integrls re used primrily s tool for omputing double integrls, where double integrl is n integrl of f (; y) over region : In this setion, we de ne double

More information

For a, b, c, d positive if a b and. ac bd. Reciprocal relations for a and b positive. If a > b then a ab > b. then

For a, b, c, d positive if a b and. ac bd. Reciprocal relations for a and b positive. If a > b then a ab > b. then Slrs-7.2-ADV-.7 Improper Definite Integrls 27.. D.dox Pge of Improper Definite Integrls Before we strt the min topi we present relevnt lger nd it review. See Appendix J for more lger review. Inequlities:

More information

ON THE C-INTEGRAL BENEDETTO BONGIORNO

ON THE C-INTEGRAL BENEDETTO BONGIORNO ON THE C-INTEGRAL BENEDETTO BONGIORNO Let F : [, b] R be differentible function nd let f be its derivtive. The problem of recovering F from f is clled problem of primitives. In 1912, the problem of primitives

More information

7.2 The Definite Integral

7.2 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 information

Green s Theorem. (2x e y ) da. (2x e y ) dx dy. x 2 xe y. (1 e y ) dy. y=1. = y e y. y=0. = 2 e

Green s Theorem. (2x e y ) da. (2x e y ) dx dy. x 2 xe y. (1 e y ) dy. y=1. = y e y. y=0. = 2 e Green s Theorem. Let be the boundry of the unit squre, y, oriented ounterlokwise, nd let F be the vetor field F, y e y +, 2 y. Find F d r. Solution. Let s write P, y e y + nd Q, y 2 y, so tht F P, Q. Let

More information

AP CALCULUS Test #6: Unit #6 Basic Integration and Applications

AP CALCULUS Test #6: Unit #6 Basic Integration and Applications AP CALCULUS Test #6: Unit #6 Bsi Integrtion nd Applitions A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION. () The ext numeril vlue of the orret

More information

Definite integral. Mathematics FRDIS MENDELU

Definite 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 information

Math 360: A primitive integral and elementary functions

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:

More information

1 i n x i x i 1. Note that kqk kp k. In addition, if P and Q are partition of [a, b], P Q is finer than both P and Q.

1 i n x i x i 1. Note that kqk kp k. In addition, if P and Q are partition of [a, b], P Q is finer than both P and Q. Chpter 6 Integrtion In this chpter we define the integrl. Intuitively, it should be the re under curve. Not surprisingly, fter mny exmples, counter exmples, exceptions, generliztions, the concept of the

More information

Definite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30

Definite 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 information

The Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.

The 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 information

Sections 5.2: The Definite Integral

Sections 5.2: The Definite Integral Sections 5.2: The Definite Integrl In this section we shll formlize the ides from the lst section to functions in generl. We strt with forml definition.. The Definite Integrl Definition.. Suppose f(x)

More information

Lecture 3. Limits of Functions and Continuity

Lecture 3. Limits of Functions and Continuity Lecture 3 Limits of Functions nd Continuity Audrey Terrs April 26, 21 1 Limits of Functions Notes I m skipping the lst section of Chpter 6 of Lng; the section bout open nd closed sets We cn probbly live

More information

1. On some properties of definite integrals. We prove

1. On some properties of definite integrals. We prove This short collection of notes is intended to complement the textbook Anlisi Mtemtic 2 by Crl Mdern, published by Città Studi Editore, [M]. We refer to [M] for nottion nd the logicl stremline of the rguments.

More information

arxiv:math/ v2 [math.ho] 16 Dec 2003

arxiv:math/ v2 [math.ho] 16 Dec 2003 rxiv:mth/0312293v2 [mth.ho] 16 Dec 2003 Clssicl Lebesgue Integrtion Theorems for the Riemnn Integrl Josh Isrlowitz 244 Ridge Rd. Rutherford, NJ 07070 jbi2@njit.edu Februry 1, 2008 Abstrct In this pper,

More information

Applications of Definite Integral

Applications of Definite Integral Chpter 5 Applitions of Definite Integrl 5.1 Are Between Two Curves In this setion we use integrls to find res of regions tht lie between the grphs of two funtions. Consider the region tht lies between

More information

Applications of Definite Integral

Applications of Definite Integral Chpter 5 Applitions of Definite Integrl 5.1 Are Between Two Curves In this setion we use integrls to find res of regions tht lie between the grphs of two funtions. Consider the region tht lies between

More information

MATH 144: Business Calculus Final Review

MATH 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 information

Section 3.6. Definite Integrals

Section 3.6. Definite Integrals The Clulus of Funtions of Severl Vribles Setion.6 efinite Integrls We will first define the definite integrl for funtion f : R R nd lter indite how the definition my be extended to funtions of three or

More information

Review on Integration (Secs ) Review: Sec Origins of Calculus. Riemann Sums. New functions from old ones.

Review on Integration (Secs ) Review: Sec Origins of Calculus. Riemann Sums. New functions from old ones. Mth 20B Integrl Clculus Lecture Review on Integrtion (Secs. 5. - 5.3) Remrks on the course. Slide Review: Sec. 5.-5.3 Origins of Clculus. Riemnn Sums. New functions from old ones. A mthemticl description

More information

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).

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). 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

More information

Solutions to Assignment 1

Solutions to Assignment 1 MTHE 237 Fll 2015 Solutions to Assignment 1 Problem 1 Find the order of the differentil eqution: t d3 y dt 3 +t2 y = os(t. Is the differentil eqution liner? Is the eqution homogeneous? b Repet the bove

More information