Definite integral. Mathematics FRDIS MENDELU


 Emory Bishop
 3 years ago
 Views:
Transcription
1 Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1
2 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 region bounded by y = f(x), the xxis nd the lines x = nd x = b? The re cn be pproximted with the sum of rectngles: We cut the intervl [, b] into subintervls. Ech of these subintervls forms the bse of rectngle, where the height of the rectngle is equl to the vlue of the function f evluted t n rbitrry point from the given subintervl. The pproximtion improves s the rectngles become nrrower nd the number of rectngles increses. We define the re of the region to be the limit of the rectngle re sums s the rectngles become smller nd smller nd the number of rectngles we use pproches infinity. Such limit cn be defined even for more generl functions nd we cll it definite integrl. The definite integrls cn be defined in mny different wys, we will define the Riemnn definite integrl. Construction of the Riemnn integrl Let f be function defined on n intervl [, b] nd suppose tht f is bounded on this intervl. The sequence of points D = {x, x 1, x,..., x n } such tht = x < x 1 < x < < x n = b is sid to be prtition of the intervl [, b]. The intervls [x, x 1 ], [x 1, x ],..., [x n 1, x n ] re clled subintervls of the prtition. The number ν(d) = mx{x i x i 1, i = 1,,..., n} is clled norm of the prtition D, i.e., the norm of the prtition is the length of the longest subintervl of the prtition. We choose n rbitrry number from ech of the subintervls ξ 1 [x, x 1 ], ξ [x 1, x ],..., ξ n [x n 1, x n ] nd we denote Ξ = {ξ 1, ξ,..., ξ n } the set of these numbers. Then the sum σ(f, D, Ξ) = n f(ξ i )(x i x i 1 ) i=1 is clled the integrl sum ssocited to the function f, the prtition D nd the choice of the numbers ξ i in Ξ. 1
3 Integrl sum: ξ 1 ξ ξ ξ 4 ξ 5 ξ 6 = x x 1 x x x 4 x 5 x 6 = b σ(f, D, Ξ) = f(ξ 1 )(x 1 x ) + f(ξ )(x x 1 ) + f(ξ )(x x ) +f(ξ 4 )(x 4 x ) + f(ξ 5 )(x 5 x 4 ) + f(ξ 6 )(x 6 x 5 ) 6 = f(ξ i )(x i x i 1 ) i=1 Refinement of the prtition: ξ 1 ξ ξ ξ n = x x 1 x x n 1 x n = b σ(f, D, Ξ) = f(ξ 1 )(x 1 x ) + f(ξ )(x x 1 ) + + f(ξ n )(x n x n 1 ) n = f(ξ i )(x i x i 1 ) i=1
4 Definition (Riemnn integrl) Let f be function defined nd bounded on n intervl [, b]. Let D 1, D, D,..., D n,... be sequence of prtitions of [, b] which stisfies lim n ν(d n) = nd Ξ 1, Ξ, Ξ,..., Ξ n,... be sequence of the corresponding choices of numbers ξ i from subintervls of the prtitions. The function f is sid to be integrble on [, b] (in sense of Riemnn) if there exists number I R with the property lim σ(f, D n, Ξ n ) = I n for every sequence of prtitions (with the bove given property) nd for rbitrry prticulr choice of the points ξ i in Ξ n. The number I is sid to be Riemnn integrl of the function f on [, b] nd it is denoted I = f(x) dx. The number is clled lower limit of the integrl nd the number b is clled n upper limit of the integrl. We hve to distinguish the definite integrls from the indefinite integrls: Indefinite integrl is set of functions. Definite integrl is limit (number). We will see, tht there is connection between the definite nd indefinite integrls, since definite integrls cn be evluted using indefinite integrls.
5 Theorem (Sufficient conditions for integrbility) Let f be function which stisfies t lest one of the following conditions: 1 f is continuous on [, b], f is monotone on [, b], f is bounded on [, b] nd contins t most finite number of discontinuities on this intervl. Then the function f is integrble (in sense of Riemnn) on [, b], i.e., f(x) dx exists. Properties of the Riemnn integrl Theorem (Additivity nd homogenity with respect to the integrnd) Let f nd g be functions which re integrble on [, b], c R. Then the functions f + g nd cf re lso integrble on [, b] nd it holds: [f(x) + g(x)] dx = cf(x) dx = c f(x) dx + f(x) dx g(x) dx Theorem (Additivity with respect to the domin of integrtion) Let f be function defined of [, b], nd let c (, b) be ny number. Then the function f is integrble on [, b] if nd only if it is integrble on both the intervls [, c] nd [c, b] nd it holds: f(x) dx = c f(x) dx + c f(x) dx 4
6 Theorem Let f nd g be functions integrble on [, b] such tht f(x) g(x) on this intervl. Then It follows from the lst theorem tht f(x) dx g(x) dx. if g(x), then g(x) dx, i.e., integrl of the nonnegtive function is nonnegtive. Evlution of the Riemnn integrl Theorem (Newton  Leibniz formul) Let f be function integrble on [, b] nd let F be n ntiderivtive of f on (, b) which is continuous on [, b]. Then f(x) dx = [F (x)] b = F (b) F (). The previous theorem sys tht to clculte the Riemnn integrl of f over [, b], ll we need to do is: 1 find n ntiderivtive F of f, clculte the number F (b) F (). 5
7 Exmple 1 x dx = [ x ] = = 19. π sin x dx = [ cos x ] π = cos π + cos = 1. 1 x dx = ( x) dx + 1 x dx = ] [ x [ x + ] 1 = = 5. Theorem (Integrtion by prts) Let u, v be functions hving continuous derivtives on [, b]. Then u(x)v (x) dx = [u(x)v(x)] b u (x)v(x) dx. Exmple 1 x ln x dx = u = ln x u = 1 x v = x v = x [ ] x = ln x x x dx [ ] x = 4 ln 1 ln 1 1 = ln 1 [ 4 1 ] 1 x dx = ln 1 = ln
8 Theorem (Substitution method) Let f be function continuous on [, b] nd let ϕ be function which hs continuous derivtive ϕ on [α, β]. Further suppose tht ϕ(x) b for x [α, β]. Then β α f[ϕ(x)]ϕ (x) dx = ϕ(β) ϕ(α) f(t) dt. The formul in the theorem cn be used from left to right (the 1 st substitution method) nd from right to left (the nd substitution method). It my hppen in some prticulr cses tht the lower limit the upper limit fter the substitution. For this reson we introduce the following extension: Extension The symbol f(x) dx cn be extended for b s follows: f(x) dx =, f(x) dx = f(x) dx. b We hve two possibilities when evluting the definite integrl with using the substitution method: 1 We use the previous theorem, i.e., we trnsform the limits of the integrl nd then we use the NewtonLeibniz formul with the new limits. (We do not substitute the originl vrible into the ntiderivtive obtined fter the integrtion.) We do not use the previous theorem. We evlute the indefinite integrl (i.e., we substitute the originl vrible fter integrtion) nd then we pply the NewtonLeibniz formul with the originl limits. 7
9 Exmple Evlute π sin x cos x dx. 1 We trnsform the limits: π sin x cos x dx = t = sin x dt = cos x dx t 1 = sin = t = sin π = 1 = 1 t dt = [ t ] 1 = 1. We do not trnsform the limits: sin x cos x dx = t = sin x dt = cos x dx = Tedy t dt = t = sin x + c. π sin x cos x dx = [ sin x ] π = sin π sin = 1. Applictions of the Riemnn integrl in geometry The re under curve nd between two curves Let f be nonnegtive nd continuous function on [, b]. The re S of the region in the plne bounded by y = f(x), the xxis nd the lines x = nd x = b is: S = f(x) dx Let f, g be continuous functions nd suppose f(x) g(x) for x [, b]. The re S of the region in the plne bounded by y = f(x), y = g(x) nd the lines x = nd x = b is: S = [f(x) g(x)] dx (The signs of f nd g re rbitrry.) 8
10 Volume of the solid of revolution I Let f be nonnegtive nd continuous function on [, b]. The volume V of the solid generted by revolving the region bounded by y = f(x), the xxis nd the lines x = nd x = b bout the xxis is: V = π f (x) dx Volume of the solid of revolution II Let f, g be nonnegtive continuous functions nd suppose f(x) g(x) for x [, b]. The volume V of the solid generted by revolving the region bounded by y = f(x), y = g(x) nd the lines x = nd x = b bout the xxis is: V = π [ f (x) g (x) ] dx 9
11 Mny other res nd volumes cn be clculted using the bove formuls since we cn cut the given region into severl pieces which stisfy the bove conditions. S = S 1 + S = c [f(x) h(x)] dx + c [g(x) h(x)] dx Exmple (Volume of the cone) Find the formul for volume of cone such tht the rdius of the bse is r nd the ltitude of the cone is v. Solution: If the following tringle revolves bout the xxis, we obtin the cone: V = π v ( r v x ) dx = πr v v x dx = πr v [ x ] v = πr v v = πr v 1
12 Exmple (Volume of the bll) Find the formul for the volume of bll with the rdius r. Solution: The eqution of circle with rdius r nd the center t [, ] is x + y = r. The upper hlfcircle is the grph of the function y = r x, the lower hlfcircle is the grph of the function y = r x. If the hlfcircle revolves bout the xxis, we obtin bll. If qurter of circle revolves bout the xxis, we obtin hlf of the bll. V = π r r = π [r x x (r x ) dx = π ] r = π r (r r (r x ) dx ) = 4πr Appliction of the definite integrl in physics Consider region bounded by the grphs of the functions y = f(x), y = g(x), nd the lines x =, x = b, where g(x) f(x) on, b. Suppose tht the region hs constnt density ρ. Then: Mss of the region: m = ρ [f(x) g(x)] dx Moments of force with respect to xxis, yxis: S x = 1 ρ [ f (x) g (x) ] dx, S y = ρ x [f(x) g(x)] dx Center of mss: T = [ Sy m, S ] x m 11
13 Exmple (Center of mss) Find the center of mss of the tringle given by the vertices [, ], [, ], [, ]. Suppose tht the density ρ is constnt. Mss: m = ρ Moments of force: S x = 1 ρ S y = ρ [ ] x x dx = ρ Center of mss: T 1 = S y m =, 4 9 x dx = 9 ρ x x dx = ρ [ x = ρ 9 = ρ ] = 9 ρ 9 = ρ x dx = ρ [ x T = S x m = = T = [, ] ] = ρ 9 = 6ρ Approximtion of definite integrls Numericl methods for pproximting the definite integrl the following cses: f(x) dx re used in The ntiderivtive of the function f hs no elementry formul, hence the NewtonLeibniz formul cnnot be used ( sin x x, sin x, ex x, e x,... ). A formul for the function f is not known, we hve only set of mesured vlues. 1
14 The trpezoidl rule The trpezoidl rule for the evlution definite integrl is bsed on pproximting the region between curve nd the xxis with trpezoids. = x x 1 x x x 4 = b Let f be function bounded on [, b]. To evlute f(x) dx : We cut the intervl [, b] into n subintervls [x, x 1 ], [x 1, x ],..., [x n 1, x n ], (x =, x n = b). Suppose tht the length of ech subintervl is h = b n. Denote y = f(x ), y 1 = f(x 1 ),..., y n = f(x n ). We pproximte the function f on [x i 1, x i ], (i = 1,,..., n) with the liner function pssing through [x i 1, y i 1 ], [x i, y i ]. This liner function is of the form Hence xi x i 1 f(x) dx y = y i 1 + y i y i 1 h xi x i 1 (x x i 1 ). [ y i 1 + y ] i y i 1 (x x i 1 ) h dx 1
15 xi x i 1 f(x) dx = xi [ y i 1 + y ] i y i 1 (x x i 1 ) dx x i 1 h [ y i 1 x + y ] xi i y i 1 (x x i 1 ) h x i 1 = y i 1 h + y i y i 1 h = h (y i 1 + y i ), h which (in cse when f is positive function ) is the wellknown formul for evluting n re of the trpezoid with corners [x i 1, ], [x i 1, y i 1 ], [x i, ], [x i, y i ]. Hence, f(x) dx = x1 x xn f(x) dx + f(x) dx + + f(x) dx x x 1 x n 1 h (y + y 1 ) + h (y 1 + y ) + + h (y n 1 + y n ) = h (y + y 1 + y + + y n 1 + y n ). The trpezoidl rule Let f be function bounded on [, b], nd let = x < x 1 < < x n = b be prtition of [, b] such tht the length of ech subintervl of this prtition is h = b n. Then f(x) dx h (y + y 1 + y + + y n 1 + y n ), where y = f(x 1 ), y = f(x ),..., y n = f(x n ). (We suppose tht f is defined t the points x, x 1,..., x n.) Some other rules for pproximting the definite integrls cn be used, e.g., the soclled Simpson s rule is bsed on pproximting curves with prbols insted of lines. 14
16 Using the computer lgebr systems Wolfrm Alph: Mthemticl Assistnt on Web (MAW): wood.mendelu.cz/mth/mwhtml/index.php?lng=en&form=min Exmple Using the Wolfrm Alph find the integrl π sin x dx. Solution: integrte sin x dx from x= to pi 15
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 information1 The Riemann Integral
The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re
More informationSections 5.2: The Definite Integral
Sections 5.2: The Definite Integrl In this section we shll formlize the ides from the lst section to functions in generl. We strt with forml definition.. The Definite Integrl Definition.. Suppose f(x)
More informationINTRODUCTION TO INTEGRATION
INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide
More informationThe Riemann Integral
Deprtment of Mthemtics King Sud University 20172018 Tble of contents 1 Antiderivtive Function nd Indefinite Integrls 2 3 4 5 Indefinite Integrls & Antiderivtive Function Definition Let f : I R be function
More informationIndefinite Integral. Chapter Integration  reverse of differentiation
Chpter Indefinite Integrl Most of the mthemticl opertions hve inverse opertions. The inverse opertion of differentition is clled integrtion. For exmple, describing process t the given moment knowing the
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationWeek 10: Riemann integral and its properties
Clculus nd Liner Algebr for Biomedicl Engineering Week 10: Riemnn integrl nd its properties H. Führ, Lehrstuhl A für Mthemtik, RWTH Achen, WS 07 Motivtion: Computing flow from flow rtes 1 We observe the
More information7.2 Riemann Integrable Functions
7.2 Riemnn Integrble Functions Theorem 1. If f : [, b] R is step function, then f R[, b]. Theorem 2. If f : [, b] R is continuous on [, b], then f R[, b]. Theorem 3. If f : [, b] R is bounded nd continuous
More informationIntegrals  Motivation
Integrls  Motivtion When we looked t function s rte of chnge If f(x) is liner, the nswer is esy slope If f(x) is nonliner, we hd to work hrd limits derivtive A relted question is the re under f(x) (but
More informationa < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1
Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the
More informationSection 6.1 Definite Integral
Section 6.1 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 e determined
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More informationChapter 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
More informationCalculus and linear algebra for biomedical engineering Week 11: The Riemann integral and its properties
Clculus nd liner lgebr for biomedicl engineering Week 11: The Riemnn integrl nd its properties Hrtmut Führ fuehr@mth.rwthchen.de Lehrstuhl A für Mthemtik, RWTH Achen Jnury 9, 2009 Overview 1 Motivtion:
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationSuppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = 2.
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
More informationChapter 6. Riemann Integral
Introduction to Riemnn integrl Chpter 6. Riemnn Integrl WonKwng Prk Deprtment of Mthemtics, The College of Nturl Sciences Kookmin University Second semester, 2015 1 / 41 Introduction to Riemnn integrl
More informationn f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1
The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the
More informationWe 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:
More informationMath 554 Integration
Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums  1 Riemnn
More informationMath& 152 Section Integration by Parts
Mth& 5 Section 7.  Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible
More informationAPPLICATIONS OF THE DEFINITE INTEGRAL
APPLICATIONS OF THE DEFINITE INTEGRAL. Volume: Slicing, disks nd wshers.. Volumes by Slicing. Suppose solid object hs boundries extending from x =, to x = b, nd tht its crosssection in plne pssing through
More informationIntegration Techniques
Integrtion Techniques. Integrtion of Trigonometric Functions Exmple. Evlute cos x. Recll tht cos x = cos x. Hence, cos x Exmple. Evlute = ( + cos x) = (x + sin x) + C = x + 4 sin x + C. cos 3 x. Let u
More informationDefinite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +
Definite Integrls 5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the lefthnd
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationThe Fundamental Theorem of Calculus
The Fundmentl Theorem of Clculus MATH 151 Clculus for Mngement J. Robert Buchnn Deprtment of Mthemtics Fll 2018 Objectives Define nd evlute definite integrls using the concept of re. Evlute definite integrls
More informationTest 3 Review. Jiwen He. I will replace your lowest test score with the percentage grade from the final exam (provided it is higher).
Test 3 Review Jiwen He Test 3 Test 3: Dec. 46 in CASA Mteril  Through 6.3. No Homework (Thnksgiving) No homework this week! Hve GREAT Thnksgiving! Finl Exm Finl Exm: Dec. 1417 in CASA You Might Be Interested
More informationf(x)dx . Show that there 1, 0 < x 1 does not exist a differentiable function g : [ 1, 1] R such that g (x) = f(x) for all
3 Definite Integrl 3.1 Introduction In school one comes cross the definition of the integrl of rel vlued function defined on closed nd bounded intervl [, b] between the limits nd b, i.e., f(x)dx s the
More informationThe Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.
Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F
More information38 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 xxis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These
More informationNumerical Analysis: Trapezoidal and Simpson s Rule
nd Simpson s Mthemticl question we re interested in numericlly nswering How to we evlute I = f (x) dx? Clculus tells us tht if F(x) is the ntiderivtive of function f (x) on the intervl [, b], then I =
More informationNUMERICAL INTEGRATION
NUMERICAL INTEGRATION How do we evlute I = f (x) dx By the fundmentl theorem of clculus, if F (x) is n ntiderivtive of f (x), then I = f (x) dx = F (x) b = F (b) F () However, in prctice most integrls
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 200910 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationZ 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.
More informationAPPROXIMATE INTEGRATION
APPROXIMATE INTEGRATION. Introduction We hve seen tht there re functions whose ntiderivtives cnnot be expressed in closed form. For these resons ny definite integrl involving these integrnds cnnot be
More informationChapter 6 Notes, Larson/Hostetler 3e
Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn
More informationThe area under the graph of f and above the xaxis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the xxis etween nd is denoted y f(x) dx nd clled the
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More information5.2 Volumes: Disks and Washers
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 crosssection or slice. In this section, we restrict
More informationAntiderivatives/Indefinite Integrals of Basic Functions
Antiderivtives/Indefinite Integrls of Bsic Functions Power Rule: In prticulr, this mens tht x n+ x n n + + C, dx = ln x + C, if n if n = x 0 dx = dx = dx = x + C nd x (lthough you won t use the second
More informationBig idea in Calculus: approximation
Big ide in Clculus: pproximtion Derivtive: f (x) = df dx f f(x +h) f(x) =, x h rte of chnge is pproximtely the rtio of chnges in the function vlue nd in the vrible in very short time Liner pproximtion:
More informationMath 131. Numerical Integration Larson Section 4.6
Mth. Numericl Integrtion Lrson Section. This section looks t couple of methods for pproimting definite integrls numericlly. The gol is to get good pproimtion of the definite integrl in problems where n
More informationFundamental Theorem of Calculus
Fundmentl Theorem of Clculus Recll tht if f is nonnegtive nd continuous on [, ], then the re under its grph etween nd is the definite integrl A= f() d Now, for in the intervl [, ], let A() e the re under
More information63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =
More informationMath 1431 Section M TH 4:00 PM 6:00 PM Susan Wheeler Office Hours: Wed 6:00 7:00 PM Online ***NOTE LABS ARE MON AND WED
Mth 43 Section 4839 M TH 4: PM 6: PM Susn Wheeler swheeler@mth.uh.edu Office Hours: Wed 6: 7: PM Online ***NOTE LABS ARE MON AND WED t :3 PM to 3: pm ONLINE Approimting the re under curve given the type
More informationNUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.
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
More informationRiemann 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 informationMA 124 January 18, Derivatives are. Integrals are.
MA 124 Jnury 18, 2018 Prof PB s oneminute introduction to clculus Derivtives re. Integrls re. In Clculus 1, we lern limits, derivtives, some pplictions of derivtives, indefinite integrls, definite integrls,
More informationImproper Integrals. Type I Improper Integrals How do we evaluate an integral such as
Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph
More information5.7 Improper Integrals
458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the
More informationMath 118: Honours Calculus II Winter, 2005 List of Theorems. L(P, f) U(Q, f). f exists for each ǫ > 0 there exists a partition P of [a, b] such that
Mth 118: Honours Clculus II Winter, 2005 List of Theorems Lemm 5.1 (Prtition Refinement): If P nd Q re prtitions of [, b] such tht Q P, then L(P, f) L(Q, f) U(Q, f) U(P, f). Lemm 5.2 (Upper Sums Bound
More informationFINALTERM EXAMINATION 2011 Calculus &. Analytical GeometryI
FINALTERM EXAMINATION 011 Clculus &. Anlyticl GeometryI Question No: 1 { Mrks: 1 )  Plese choose one If f is twice differentible function t sttionry point x 0 x 0 nd f ''(x 0 ) > 0 then f hs reltive...
More informationdifferent methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s).
Mth 1A with Professor Stnkov Worksheet, Discussion #41; Wednesdy, 12/6/217 GSI nme: Roy Zho Problems 1. Write the integrl 3 dx s limit of Riemnn sums. Write it using 2 intervls using the 1 x different
More informationImproper Integrals. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Improper Integrls MATH 2, Clculus II J. Robert Buchnn Deprtment of Mthemtics Spring 28 Definite Integrls Theorem (Fundmentl Theorem of Clculus (Prt I)) If f is continuous on [, b] then b f (x) dx = [F(x)]
More informationF (x) dx = F (x)+c = u + C = du,
35. The Substitution Rule An indefinite integrl of the derivtive F (x) is the function F (x) itself. Let u = F (x), where u is new vrible defined s differentible function of x. Consider the differentil
More informationl 2 p2 n 4n 2, the total surface area of the
Week 6 Lectures Sections 7.5, 7.6 Section 7.5: Surfce re of Revolution Surfce re of Cone: Let C be circle of rdius r. Let P n be n nsided regulr polygon of perimeter p n with vertices on C. Form cone
More informationWe partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.
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
More informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More informationMidpoint Approximation
Midpoint Approximtion Sometimes, we need to pproximte n integrl of the form R b f (x)dx nd we cnnot find n ntiderivtive in order to evlute the integrl. Also we my need to evlute R b f (x)dx where we do
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationFor a continuous function f : [a; b]! R we wish to define the Riemann integral
Supplementry Notes for MM509 Topology II 2. The Riemnn Integrl Andrew Swnn For continuous function f : [; b]! R we wish to define the Riemnn integrl R b f (x) dx nd estblish some of its properties. This
More informationHomework 4. (1) If f R[a, b], show that f 3 R[a, b]. If f + (x) = max{f(x), 0}, is f + R[a, b]? Justify your answer.
Homework 4 (1) If f R[, b], show tht f 3 R[, b]. If f + (x) = mx{f(x), 0}, is f + R[, b]? Justify your nswer. (2) Let f be continuous function on [, b] tht is strictly positive except finitely mny points
More informationf(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral
Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one
More informationMath 0230 Calculus 2 Lectures
Mth Clculus Lectures Chpter 7 Applictions of Integrtion Numertion of sections corresponds to the text Jmes Stewrt, Essentil Clculus, Erly Trnscendentls, Second edition. Section 7. Ares Between Curves Two
More informationTHE EXISTENCEUNIQUENESS THEOREM FOR FIRSTORDER DIFFERENTIAL EQUATIONS.
THE EXISTENCEUNIQUENESS THEOREM FOR FIRSTORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrdlindeloftheorem/ This document is proof of the existenceuniqueness theorem
More informationReview of Riemann Integral
1 Review of Riemnn Integrl In this chpter we review the definition of Riemnn integrl of bounded function f : [, b] R, nd point out its limittions so s to be convinced of the necessity of more generl integrl.
More informationChapter 5 : Continuous Random Variables
STAT/MATH 395 A  PROBABILITY II UW Winter Qurter 216 Néhémy Lim Chpter 5 : Continuous Rndom Vribles Nottions. N {, 1, 2,...}, set of nturl numbers (i.e. ll nonnegtive integers); N {1, 2,...}, set of ll
More informationMATH , Calculus 2, Fall 2018
MATH 362, 363 Clculus 2, Fll 28 The FUNdmentl Theorem of Clculus Sections 5.4 nd 5.5 This worksheet focuses on the most importnt theorem in clculus. In fct, the Fundmentl Theorem of Clculus (FTC is rgubly
More informationLecture 3 ( ) (translated and slightly adapted from lecture notes by Martin Klazar)
Lecture 3 (5.3.2018) (trnslted nd slightly dpted from lecture notes by Mrtin Klzr) Riemnn integrl Now we define precisely the concept of the re, in prticulr, the re of figure U(, b, f) under the grph of
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationSection Areas and Distances. Example 1: Suppose a car travels at a constant 50 miles per hour for 2 hours. What is the total distance traveled?
Section 5.  Ares nd Distnces Exmple : Suppose cr trvels t constnt 5 miles per hour for 2 hours. Wht is the totl distnce trveled? Exmple 2: Suppose cr trvels 75 miles per hour for the first hour, 7 miles
More information6.5 Numerical Approximations of Definite Integrals
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,
More informationReview on Integration (Secs ) Review: Sec Origins of Calculus. Riemann Sums. New functions from old ones.
Mth 20B Integrl Clculus Lecture Review on Integrtion (Secs. 5.  5.3) Remrks on the course. Slide Review: Sec. 5.5.3 Origins of Clculus. Riemnn Sums. New functions from old ones. A mthemticl description
More informationSection 4.8. D v(t j 1 ) t. (4.8.1) j=1
Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions
More informationTest , 8.2, 8.4 (density only), 8.5 (work only), 9.1, 9.2 and 9.3 related test 1 material and material from prior classes
Test 2 8., 8.2, 8.4 (density only), 8.5 (work only), 9., 9.2 nd 9.3 relted test mteril nd mteril from prior clsses Locl to Globl Perspectives Anlyze smll pieces to understnd the big picture. Exmples: numericl
More informationRiemann Integrals and the Fundamental Theorem of Calculus
Riemnn Integrls nd the Fundmentl Theorem of Clculus Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University September 16, 2013 Outline Grphing Riemnn Sums
More informationCalculus III Review Sheet
Clculus III Review Sheet 1 Definitions 1.1 Functions A function is f is incresing on n intervl if x y implies f(x) f(y), nd decresing if x y implies f(x) f(y). It is clled monotonic if it is either incresing
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More informationWe know that if f is a continuous nonnegative function on the interval [a, b], then b
1 Ares Between Curves c 22 Donld Kreider nd Dwight Lhr We know tht if f is continuous nonnegtive function on the intervl [, b], then f(x) dx is the re under the grph of f nd bove the intervl. We re going
More informationLECTURE. 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 informationMath Calculus with Analytic Geometry II
orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove xxis) ( bove f under xxis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem
More informationCalculus II: Integrations and Series
Clculus II: Integrtions nd Series August 7, 200 Integrls Suppose we hve generl function y = f(x) For simplicity, let f(x) > 0 nd f(x) continuous Denote F (x) = re under the grph of f in the intervl [,x]
More informationMath 120 Answers for Homework 13
Mth 12 Answers for Homework 13 1. In this problem we will use the fct tht if m f(x M on n intervl [, b] (nd if f is integrble on [, b] then (* m(b f dx M(b. ( The function f(x = 1 + x 3 is n incresing
More informationSpace Curves. Recall the parametric equations of a curve in xyplane and compare them with parametric equations of a curve in space.
Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xyplne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)
More informationSection 5.4 Fundamental Theorem of Calculus 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus 1
Section 5.4 Fundmentl Theorem of Clculus 2 Lectures College of Science MATHS : Clculus (University of Bhrin) Integrls / 24 Definite Integrl Recll: The integrl is used to find re under the curve over n
More informationLecture 19: Continuous Least Squares Approximation
Lecture 19: Continuous Lest Squres Approximtion 33 Continuous lest squres pproximtion We begn 31 with the problem of pproximting some f C[, b] with polynomil p P n t the discrete points x, x 1,, x m for
More informationP 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
More informationMAT612REAL ANALYSIS RIEMANN STIELTJES INTEGRAL
MAT612REAL ANALYSIS RIEMANN STIELTJES INTEGRAL DR. RITU AGARWAL MALVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR, INDIA302017 Tble of Contents Contents Tble of Contents 1 1. Introduction 1 2. Prtition
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationExam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH1105 Instructor: Attila Máté 1
Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixedpoint itertion to converge when solving the eqution
More information1 The fundamental theorems of calculus.
The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl new nme for ntiderivtive. Differentiting integrls. Tody we provide the connection
More informationSummer MTH142 College Calculus 2. Section J. Lecture Notes. Yin Su University at Buffalo
Summer 6 MTH4 College Clculus Section J Lecture Notes Yin Su University t Bufflo yinsu@bufflo.edu Contents Bsic techniques of integrtion 3. Antiderivtive nd indefinite integrls..............................................
More information1 Part II: Numerical Integration
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
More information4.4 Areas, Integrals and Antiderivatives
. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order
More informationMath 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8
Mth 3 Fll 0 The scope of the finl exm will include: Finl Exm Review. Integrls Chpter 5 including sections 5. 5.7, 5.0. Applictions of Integrtion Chpter 6 including sections 6. 6.5 nd section 6.8 3. Infinite
More informationCMDA 4604: Intermediate Topics in Mathematical Modeling Lecture 19: Interpolation and Quadrature
CMDA 4604: Intermedite Topics in Mthemticl Modeling Lecture 19: Interpoltion nd Qudrture In this lecture we mke brief diversion into the res of interpoltion nd qudrture. Given function f C[, b], we sy
More informationContinuous Random Variables
STAT/MATH 395 A  PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is relvlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More informationThe problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests.
ADVANCED CALCULUS PRACTICE PROBLEMS JAMES KEESLING The problems tht follow illustrte the methods covered in clss. They re typicl of the types of problems tht will be on the tests. 1. Riemnn Integrtion
More information