Calculus II - Simon Fraser University

1 Calculus IIIntegral CalculusLecture NotesVeselin Jungic & Jamie MulhollandDepartment of MathematicsSimon Fraser Universityc Draft date January 2, 2018 ContentsContentsiPrefaceiiiGreek Alphabetv1 Areas and Distances .. The Definite Integral .. The Fundamental Theorem of Calculus .. Indefinite Integrals .. The Substitution Rule ..272 Applications of Areas Between Curves .. Areas in Polar Coordinates .. Volumes .. Volumes by Cylindrical Shells ..503 Techniques of Integration By Parts .. Trigonometric Integrals .. Trigonometric Substitutions .. Integration of Rational Functions by Partial Fractions .. Strategy for Integration .. Approximate Integration .. Improper Integrals ..894 Further Applications of Arc Length .. Area of a Surface of Revolution .. Calculus with Parametric Curves .. 1045 Infinite Sequences and Sequences .. Series .. The Integral Test and Estimates of Sums.

2 The Comparison Test .. Alternating Series .. Absolute Convergence and the Ratio and Root Test .. Strategy for Testing Series .. Power Series .. Representation of Functions as Power Series .. Taylor and Maclaurin Series .. Applications of Taylor Polynomials .. 1596 A First Look at Differential Modeling with Differential Equations, Direction Fields .. Separable Equations .. Models for Population Growth .. 1767 Review Midterm 1 Review Package .. Midterm 2 Review Package .. Final Exam Practice Questions .. 196 Bibliography201 Index202 PrefaceThis booklet contains our notes for coursesMath 152 - Calculus IIat Simon Fraser University . Studentsare expected to bring this booklet to each lecture and to follow along, filling in the details in the blanksprovided, during the of terms are stated inorange boxes and theorems appear inblue boxes .Next to some examples you ll see [link to applet].

3 The link will take you to an online interactive applet toaccompany the example - just like the ones used by your instructor in the lecture. Clicking the link abovewill take you to the following website containing all the applets: jtmulhol/ Calculus -applets/ it to some section headings you ll notice a QR code. They look like theimage on the one provides a link to a webpage (could be a youtube video, or accessto online Sage code). For example this one takes you to the Wikipediapage which explains what a QR code is. Use a QR code scanner on yourphone or tablet and it will quickly take you off to the webpage. The app Red Laser is a good QR code scanner which is available for free (iphone,android, windows phone).If you don t have a scanner, don t worry, I ve hyperlinked all the QR codes so if you are viewing thisdocument electronically then you can just click on the image. However, if you are viewing a printed versionthen this is where the scanner comes in handy, but again if you don t have one you can manually type inthe url that is provided below the offer a special thank you to Keshav Mukunda for his many contributions to these project such as this can be free from errors and incompleteness.

4 We will be grateful to everyonewho points out any typos, incorrect statements, or sends any other suggestion on how to improve Fraser UniversityJanuary 2, 2018iiiivGreek Alphabetlowercasecapitalnamepronunciatio nlowercasecapitalnamepronunciation Aalpha(al-fah) Nnu(new) Bbeta(bay-tah) xi(zie) gamma(gam-ah)o Oomicron(om-e-cron) delta(del-ta) pi(pie) Eepsilon(ep-si-lon) Prho(roe) Zzeta(zay-tah) sigma(sig-mah) Heta(ay-tah) Ttau(taw) theta(thay-tah) upsilon(up-si-lon) Iiota(eye-o-tah) phi(fie) Kkappa(cap-pah) Xchi(kie) lambda(lamb-dah) psi(si) Mmu(mew) omega(oh-may-gah)vPart 1 Integrals1 PART1: Areas and Distances(This lecture corresponds to Section of Stewart sCalculus.) can never know for sure what a desertedarealooks like.(George Carlin, American stand-up Comedian, Actor and Author, 1937-2008) is the meaning of the wordarea? dictionary:areanoun(a) a particular part of a place, piece of land or country;(b) the size of a flat surface calculated by multiplying its length by its width;(c) a subject or activity, or a part of it.

5 (d) (Wikipedia) - Area is a physical quantity expressing the sizeof a part of a the area of the region in the coordinate plane bounded by the coordinate axes andlinesx= 2andy= the area of the region in the coordinate plane bounded by thex-axis and linesy= 2xandx= the area of the region in the coordinate plane bounded by thex-axis and linesy=x2andx= : area of the region in the coordinate plane bounded by thex-axis and linesy=x2andx= : (Over- and under-estimates.)In the previous example, show thatlimn Rn= 9andlimn Ln= more general : A functionfthat is continuous on a closed interval[a,b].Letn N, and define x=b + xx2=a+ 2 xx3=a+ 3 +n x= (x1) x+f(x2) x+..+f(xn) x.( R stands for right-hand , since we are using the right hand endpoints of the little rectangles.) of the regionSthat lies under the graph of the continuous functionfover and interval[a,b]is the limit of the sum of the areas of approximating rectanglesRn. That is,A= limn Rn= limn [f(x1) +f(x2) +.]

6 +f(xn)] more compactsigma notationcan be used to write this asA= limn Rn= limn (n i=1f(xi)) : the area under the graph off(x) = 100 3x2fromx= 1tox= the definition of area, we haveA= limn (n i=1f(xi)) the distance traveled by an object during a certain time period if thevelocity of the object is known at all = velocity timePART1: ANDDISTANCES613. Additional NotesPART1: The Definite Integral(This lecture corresponds to Section of Stewart sCalculus.) After years of finding mathematics easy, I finally reached integral Calculus and came upagainst a barrier. I realized that this was as far as I could go, and to this day I have never successfullygone beyond it in any but the most superficial way. (Isaac Asimov, Russian-born American author and biochemist, best known for his works of sciencefiction, 1920-1992) Definite a continuous function defined on the closed interval[a,b], wedivide[a,b]intonsubintervals of equal width x= (b a)/n.

7 Letx0=a, x1, x2, .., xn=bbe the end points of these subintervals. Letx 1,x 2,..,x nbe anysample pointsin these subintervals, sox ilies in theith subinterval[xi 1,xi].Then thedefinite integral offfromatobis written as baf(x)dx,and is defined as follows: baf(x)dx= limn n i=1f(x i) xPART1: definite integral: some terminology baf(x)dx= limn n i=1f(x i) x is theintegral sign f(x)is theintegrand aandbare thelimits of integration: a-lower limit b-upper limit The procedure of calculating an integral is calledintegration. n i=1f(x i) xis called aRiemann sum(named after the German mathematician Bernhard Riemann,1826-1866) Facts.(a) Iff(x)>0on[a,b]then baf(x)dx > (x)<0on[a,b]then baf(x)dx <0.(b) For a general functionf, baf(x)dx=(signed area of the region) = (area abovex-axis) - (area belowx-axis)(c) For every >0there exists a numbern Nsuch that baf(x)dx n i=1f(x i) x < for everyn > Nand every choice ofx 1,x 2,..,x n.(d) Letfbe continuous on[a,b]and leta=x0< x1< x2<.

8 < xn=bbe any partition of[a,b]. Let xi=xi xi 1, and supposemax xiapproaches0asntends to infinity. Then baf(x)dx= limn n i=1f(x i) xiPART1: facts you just have to (a)n i=1i=n(n+ 1)2(b)n i=1i2=n(n+ 1)(2n+ 1)6(c)n i=1i3=(n(n+ 1)2)2(d)n i=1c=cn(e)n i=1(cai) =cn i=1ai(f)n i=1(ai bi) =n i=1ai n i=1bi1 For visual proofs of (a) and (b) see Goldoni, G. (2002).A visual proof for the sum of the first n squares and for the sum of the firstn factorials of order two. The Mathematical Intelligencer 24 (4): 6769. You can access the Mathematical Intelligencer through theSFU Library web site: : 20(x2 x) the limitlimn n i=1(1 +xi) cosxi xas a definite integral on the interval[ ,2 ]. 20 4 x2dx= .PART1: a good sample point ..Midpoint approximate an integral it is usually better to choosex ito be the midpointxiofthe interval[xi 1,xi]: baf(x)dx n i=1f(xi) x= x[f(x1) +f(x2) +..+f(xn)]Recall the midpoint of an interval[xi 1,xi]is given byxi=12(xi 1+xi). the Midpoint Rule withn= 4to approximate the integral : Special Properties of the Integral.

9 (a) Ifa > bthen baf(x)dx= abf(x)dx.(b) Ifa=bthen baf(x)dx= More Properties of the Integral.(a) Ifcis a constant, then bacdx=c(b a)(b) ba[f(x) g(x)]dx= baf(x)dx bag(x)dx(c) Ifcis a constant, then bacf(x)dx=c baf(x)dx(d) caf(x)dx+ bcf(x)dx= baf(x) 30(2x 3 9 x2) : 30f(x)dxiff(x) ={1 xifx [0,1] 1 (x 2)2ifx (1,3] Properties of the definite integral.(a) Iff(x) 0fora x b, then baf(x)dx 0.(b) Iff(x) g(x)fora x b, then baf(x)dx bag(x)dx.(c) IfmandMare constants, andm f(x) Mfora x b, thenm(b a) baf(x)dx M(b a)PART1: 21e x2dx (a) Iffis continuous on[a,b], show that baf(x)dx ba|f(x)|dx.(b) Show that iffis continuous on[0,2 ]then 2 0f(x) sin 2xdx 2 0|f(x)| : Additional NotesPART1: The Fundamental Theorem of Calculus (This lecture corresponds to Section of Stewart sCalculus.) All of my fundamental principles that were instilled in me in my home, from my childhood,are still with me. (Hakeem Abdul Olajuwon, a former NBA player,1963-) every continuous functionfhave an antiderivative?)}

10 That is, does there exist afunctionFsuch thatF (x) =f(x)? is the antiderivative off(x) =sinxx? Fundamental Theorem of Calculus , Part a continuous on[a,b], then the functiongdefined byg(x) = xaf(t)dt, a x bis continuous on[a,b]and differentiable on(a,b), andg (x) =f(x).PART1: the Fundamental Theorem of Calculus , Part 1, to find the derivative of the followingfunctions:(a)g(x) = x1sinttdt(b)g(x) = x20sint dt(c)g(x) = h(x)0f(t)dt(d)g(x) = ex 3xln(1 +t2)dtPART1: Fundamental Theorem of Calculus , Part continuous on[a,b], then baf(x)dx=F(b) F(a)whereFis any antiderivative off. That is, a function such thatF = : the following integrals:(a) 10xdx(b) 32exdx(c) 0sinx dx(d) 10dx1 +x2 PART1: Piecewise (x) = 0ifx <0xif0 x 12 xif1< x 20ifx >2and letg(x) = x0f(t)dt.(a) Find an expression forg(x)similar to the one forf(x).(b) Sketch the graphs offandg.(c) Where isfdifferentiable? Where isgdifferentiable?PART1: OFCALCULUS219. Additional NotesPART1: Indefinite Integrals(This lecture corresponds to Section of Stewart sCalculus.)