109301798-Calculus-2.pdf - CALCULUS II Paul Dawkins...

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Unformatted text preview: CALCULUS II Paul Dawkins Calculus II Table of Contents Preface............................................................................................................................................ iii Outline............................................................................................................................................. v Integration Techniques .................................................................................................................. 1 Introduction ................................................................................................................................................ 1 Integration by Parts .................................................................................................................................... 3 Integrals Involving Trig Functions ........................................................................................................... 13 Trig Substitutions ..................................................................................................................................... 23 Partial Fractions ....................................................................................................................................... 34 Integrals Involving Roots ......................................................................................................................... 42 Integrals Involving Quadratics ................................................................................................................. 44 Using Integral Tables ............................................................................................................................... 52 Integration Strategy .................................................................................................................................. 55 Improper Integrals .................................................................................................................................... 62 Comparison Test for Improper Integrals .................................................................................................. 69 Approximating Definite Integrals ............................................................................................................ 76 Applications of Integrals ............................................................................................................. 83 Introduction .............................................................................................................................................. 83 Arc Length ............................................................................................................................................... 84 Surface Area ............................................................................................................................................. 90 Center of Mass ......................................................................................................................................... 96 Hydrostatic Pressure and Force .............................................................................................................. 100 Probability .............................................................................................................................................. 105 Parametric Equations and Polar Coordinates ........................................................................ 109 Introduction ............................................................................................................................................ 109 Parametric Equations and Curves........................................................................................................... 110 Tangents with Parametric Equations ...................................................................................................... 121 Area with Parametric Equations ............................................................................................................. 128 Arc Length with Parametric Equations .................................................................................................. 131 Surface Area with Parametric Equations ................................................................................................ 135 Polar Coordinates ................................................................................................................................... 137 Tangents with Polar Coordinates ........................................................................................................... 147 Area with Polar Coordinates .................................................................................................................. 149 Arc Length with Polar Coordinates ........................................................................................................ 156 Surface Area with Polar Coordinates ..................................................................................................... 158 Arc Length and Surface Area Revisited ................................................................................................. 159 Sequences and Series ................................................................................................................. 161 Introduction ............................................................................................................................................ 161 Sequences ............................................................................................................................................... 163 More on Sequences ................................................................................................................................ 173 Series – The Basics ................................................................................................................................ 179 Series – Convergence/Divergence.......................................................................................................... 185 Series – Special Series............................................................................................................................ 194 Integral Test ........................................................................................................................................... 202 Comparison Test / Limit Comparison Test ............................................................................................ 211 Alternating Series Test ........................................................................................................................... 220 Absolute Convergence ........................................................................................................................... 226 Ratio Test ............................................................................................................................................... 230 Root Test ................................................................................................................................................ 237 Strategy for Series .................................................................................................................................. 240 Estimating the Value of a Series ............................................................................................................ 243 Power Series ........................................................................................................................................... 254 Power Series and Functions ................................................................................................................... 262 Taylor Series .......................................................................................................................................... 269 © 2007 Paul Dawkins i Calculus II Applications of Series ............................................................................................................................ 279 Binomial Series ...................................................................................................................................... 284 Vectors ........................................................................................................................................ 286 Introduction ............................................................................................................................................ 286 Vectors – The Basics .............................................................................................................................. 287 Vector Arithmetic................................................................................................................................... 291 Dot Product ............................................................................................................................................ 296 Cross Product ......................................................................................................................................... 304 Three Dimensional Space .......................................................................................................... 310 Introduction ............................................................................................................................................ 310 The 3-D Coordinate System ................................................................................................................... 312 Equations of Lines.................................................................................................................................. 318 Equations of Planes ................................................................................................................................ 324 Quadric Surfaces .................................................................................................................................... 327 Functions of Several Variables............................................................................................................... 333 Vector Functions .................................................................................................................................... 340 Calculus with Vector Functions ............................................................................................................. 349 Tangent, Normal and Binormal Vectors ................................................................................................ 352 Arc Length with Vector Functions ......................................................................................................... 356 Curvature ................................................................................................................................................ 359 Velocity and Acceleration ...................................................................................................................... 361 Cylindrical Coordinates.......................................................................................................................... 364 Spherical Coordinates ............................................................................................................................ 366 © 2007 Paul Dawkins ii Calculus II Preface Here are my online notes for my Calculus II course that I teach here at Lamar University. Despite the fact that these are my “class notes”, they should be accessible to anyone wanting to learn Calculus II or needing a refresher in some of the topics from the class. These notes do assume that the reader has a good working knowledge of Calculus I topics including limits, derivatives and basic integration and integration by substitution. Calculus II tends to be a very difficult course for many students. There are many reasons for this. The first reason is that this course does require that you have a very good working knowledge of Calculus I. The Calculus I portion of many of the problems tends to be skipped and left to the student to verify or fill in the details. If you don’t have good Calculus I skills, and you are constantly getting stuck on the Calculus I portion of the problem, you will find this course very difficult to complete. The second, and probably larger, reason many students have difficulty with Calculus II is that you will be asked to truly think in this class. That is not meant to insult anyone; it is simply an acknowledgment that you can’t just memorize a bunch of formulas and expect to pass the course as you can do in many math classes. There are formulas in this class that you will need to know, but they tend to be fairly general. You will need to understand them, how they work, and more importantly whether they can be used or not. As an example, the first topic we will look at is Integration by Parts. The integration by parts formula is very easy to remember. However, just because you’ve got it memorized doesn’t mean that you can use it. You’ll need to be able to look at an integral and realize that integration by parts can be used (which isn’t always obvious) and then decide which portions of the integral correspond to the parts in the formula (again, not always obvious). Finally, many of the problems in this course will have multiple solution techniques and so you’ll need to be able to identify all the possible techniques and then decide which will be the easiest technique to use. So, with all that out of the way let me also get a couple of warnings out of the way to my students who may be here to get a copy of what happened on a day that you missed. 1. Because I wanted to make this a fairly complete set of notes for anyone wanting to learn calculus I have included some material that I do not usually have time to cover in class and because this changes from semester to semester it is not noted here. You will need to find one of your fellow class mates to see if there is something in these notes that wasn’t covered in class. 2. In general I try to work problems in class that are different from my notes. However, with Calculus II many of the problems are difficult to make up on the spur of the moment and so in this class my class work will follow these notes fairly close as far as worked problems go. With that being said I will, on occasion, work problems off the top of my head when I can to provide more examples than just those in my notes. Also, I often © 2007 Paul Dawkins iii Calculus II don’t have time in class to work all of the problems in the notes and so you will find that some sections contain problems that weren’t worked in class due to time restrictions. 3. Sometimes questions in class will lead down paths that are not covered here. I try to anticipate as many of the questions as possible in writing these up, but the reality is that I can’t anticipate all the questions. Sometimes a very good question gets asked in class that leads to insights that I’ve not included here. You should always talk to someone who was in class on the day you missed and compare these notes to their notes and see what the differences are. 4. This is somewhat related to the previous three items, but is important enough to merit its own item. THESE NOTES ARE NOT A SUBSTITUTE FOR ATTENDING CLASS!! Using these notes as a substitute for class is liable to get you in trouble. As already noted not everything in these notes is covered in class and often material or insights not in these notes is covered in class. © 2007 Paul Dawkins iv Calculus II Outline Here is a listing and brief description of the material in this set of notes. Integration Techniques Integration by Parts – Of all the integration techniques covered in this chapter this is probably the one that students are most likely to run into down the road in other classes. Integrals Involving Trig Functions – In this section we look at integrating certain products and quotients of trig functions. Trig Substitutions – Here we will look using substitutions involving trig functions and how they can be used to simplify certain integrals. Partial Fractions – We will use partial fractions to allow us to do integrals involving some rational functions. Integrals Involving Roots – We will take a look at a substitution that can, on occasion, be used with integrals involving roots. Integrals Involving Quadratics – In this section we are going to look at some integrals that involve quadratics. Using Integral Tables – Here we look at using Integral Tables as well as relating new integrals back to integrals that we already know how to do. Integration Strategy – We give a general set of guidelines for determining how to evaluate an integral. Improper Integrals – We will look at integrals with infinite intervals of integration and integrals with discontinuous integrands in this section. Comparison Test for Improper Integrals – Here we will use the Comparison Test to determine if improper integrals converge or diverge. Approximating Definite Integrals – There are many ways to approximate the value of a definite integral. We will look at three of them in this section. Applications of Integrals Arc Length – We’ll determine the length of a curve in this section. Surface Area – In this section we’ll determine the surface area of a solid of revolution. Center of Mass – Here we will determine the center of mass or centroid of a thin plate. Hydrostatic Pressure and Force – We’ll determine the hydrostatic pressure and force on a vertical plate submerged in water. Probability – Here we will look at probability density functions and computing the mean of a probability density function. Parametric Equations and Polar Coordinates Parametric Equations and Curves – An introduction to parametric equations and parametric curves (i.e. graphs of parametric equations) Tangents with Parametric Equations – Finding tangent lines to parametric curves. Area with Parametric Equations – Finding the area under a parametric curve. © 2007 Paul Dawkins v Calculus II Arc Length with Parametric Equations – Determining the length of a parametric curve. Surface Area with Parametric Equations – Here we will determine the surface area of a solid obtained by rotating a parametric curve about an axis. Polar Coordinates – We’ll introduce polar coordinates in this section. We’ll look at converting between polar coordinates and Cartesian coordinates as well as some basic graphs in polar coordinates. Tangents with Polar Coordinates – Finding tangent lines of polar curves. Area with Polar Coordinates – Finding the area enclosed by a polar curve. Arc Length with Polar Coordinates – Determining the length of a polar curve. Surface Area with Polar Coordinates – Here we will determine the surface area of a solid obtained by rotating a polar curve about an axis. Arc Length and Surface Area Revisited – In this section we will summarize all the arc length and surface area formulas from the last two chapters. Sequences and Series Sequences – We will start the chapter off with a brief discussion of sequences. This section will focus on the basic terminology and convergence of sequences More on Sequences – Here we will take a quick look about monotonic and bounded sequences. Series – The Basics – In this section we will discuss some of the basics of infinite series. Series – Convergence/Divergence – Most of this chapter will be about the convergence/divergence of a series so we will give the basic ideas and definitions in this section. Series – Special Series – We will look at the Geometric Series, Telescoping Series, and Harmonic Series in this section. Integral Test – Using the Integral Test to determine if a series converges or diverges. Comparison Test/Limit Comparison Test – Using the Comparison Test and Limit Co...
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