Calculus II Paul Dawkins

CALCULUSII  PaulDawkins Calculus II © 2007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx 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 Calculus II © 2007 Paul Dawkins ii http://tutorial.math.lamar.edu/terms.aspx 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 355 Curvature 358 Velocity and Acceleration 360 Cylindrical Coordinates 363 Spherical Coordinates 365 Calculus II © 2007 Paul Dawkins iii http://tutorial.math.lamar.edu/terms.aspx 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 acknowledgement 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 and 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 Calculus II © 2007 Paul Dawkins iv http://tutorial.math.lamar.edu/terms.aspx 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. Calculus II © 2007 Paul Dawkins v http://tutorial.math.lamar.edu/terms.aspx 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. Calculus II © 2007 Paul Dawkins vi http://tutorial.math.lamar.edu/terms.aspx 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 Comparison Tests to determine if a series converges or diverges. Alternating Series Test – Using the Alternating Series Test to determine if a series converges or diverges. Absolute Convergence – A brief discussion on absolute convergence and how it differs from convergence. Ratio Test – Using the Ratio Test to determine if a series converges or diverges. Root Test – Using the Root Test to determine if a series converges or diverges. Strategy for Series – A set of general guidelines to use when deciding which test to use. Estimating the Value of a Series – Here we will look at estimating the value of an infinite series. Power Series – An introduction to power series and some of the basic concepts. Power Series and Functions – In this section we will start looking at how to find a power series representation of a function. Taylor Series – Here we will discuss how to find the Taylor/Maclaurin Series for a function. Applications of Series – In this section we will take a quick look at a couple of applications of series. Binomial Series – A brief look at binomial series. Vectors Calculus II © 2007 Paul Dawkins vii http://tutorial.math.lamar.edu/terms.aspx Vectors – The Basics – In this section we will introduce some of the basic concepts about vectors. Vector Arithmetic – Here we will give the basic arithmetic operations for vectors. Dot Product – We will discuss the dot product in this section as well as an application or two. Cross Product – In this section we’ll discuss the cross product and see a quick application. Three Dimensional Space This is the only chapter that exists in two places in my notes. When I originally wrote these notes all of these topics were covered in Calculus II however, we have since moved several of them into Calculus III. So, rather than split the chapter up I have kept it in the Calculus II notes and also put a copy in the Calculus III notes. The 3-D Coordinate System – We will introduce the concepts and notation for the three dimensional coordinate system in this section. Equations of Lines – In this section we will develop the various forms for the equation of lines in three dimensional space. Equations of Planes – Here we will develop the equation of a plane. Quadric Surfaces – In this section we will be looking at some examples of quadric surfaces. Functions of Several Variables – A quick review of some important topics about functions of several variables. Vector Functions – We introduce the concept of vector functions in this section. We concentrate primarily on curves in three dimensional space. We will however, touch briefly on surfaces as well. Calculus with Vector Functions – Here we will take a quick look at limits, derivatives, and integrals with vector functions. Tangent, Normal and Binormal Vectors – We will define the tangent, normal and binormal vectors in this section. Arc Length with Vector Functions – In this section we will find the arc length of a vector function. Curvature – We will determine the curvature of a function in this section. Velocity and Acceleration – In this section we will revisit a standard application of derivatives. We will look at the velocity and acceleration of an object whose position function is given by a vector function. Cylindrical Coordinates – We will define the cylindrical coordinate system in this section. The cylindrical coordinate system is an alternate coordinate system for the three dimensional coordinate system. Spherical Coordinates – In this section we will define the spherical coordinate system. The spherical coordinate system is yet another alternate coordinate system for the three dimensional coordinate system. Calculus II © 2007 Paul Dawkins 0 http://tutorial.math.lamar.edu/terms.aspx Calculus II © 2007 Paul Dawkins 1 http://tutorial.math.lamar.edu/terms.aspx IntegrationTechniques Introduction In this chapter we are going to be looking at various integration techniques. There are a fair number of them and some will be easier than others. The point of the chapter is to teach you these new techniques and so this chapter assumes that you’ve got a fairly good working knowledge of basic integration as well as substitutions with integrals. In fact, most integrals involving “simple” substitutions will not have any of the substitution work shown. It is going to be assumed that you can verify the substitution portion of the integration yourself. Also, most of the integrals done in this chapter will be indefinite integrals. It is also assumed that once you can do the indefinite integrals you can also do the definite integrals and so to conserve space we concentrate mostly on indefinite integrals. There is one exception to this and that is the Trig Substitution section and in this case there are some subtleties involved with definite integrals that we’re going to have to watch out for. Outside of that however, most sections will have at most one definite integral example and some sections will not have any definite integral examples. Here is a list of topics that are covered in this chapter. 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. [...]... Problems] © 2007 Paul Dawkins 7 http://tutorial.math.lamar.edu/terms.aspx Calculus II So, we used two different integration techniques in this example and we got two different answers The obvious question then should be : Did we do something wrong? Actually, we didn’t do anything wrong We need to remember the following fact from Calculus I If f ′ ( x ) = g ′ ( x ) then f ( x ) = g ( x ) + c In other... one secant in the integral © 2007 Paul Dawkins 18 http://tutorial.math.lamar.edu/terms.aspx Calculus II Example 7 Evaluate the following integral ∫ tan x dx Solution To do this integral all we need to do is recall the definition of tangent in terms of sine and cosine and then this integral is nothing more than a Calculus I substitution ⌠ sin x ∫ tan x dx = ⎮ cos x dx ⌡ u = cos x 1 = −⌠ du ⎮ ⌡u = − ln... they would just end up absorbing this one Finally, rewrite the formula as follows and we arrive that the integration by parts formula © 2007 Paul Dawkins 3 http://tutorial.math.lamar.edu/terms.aspx Calculus II ∫ f g ′ dx = fg − ∫ f ′ g dx This is not the easiest formula to use however So, let’s do a couple of substitutions u = f ( x) v = g ( x) du = f ′ ( x ) dx dv = g ′ ( x ) dx Both of these are just... e6 x + c 6 36 Once we have done the last integral in the problem we will add in the constant of integration to get our final answer © 2007 Paul Dawkins 4 http://tutorial.math.lamar.edu/terms.aspx Calculus II Next, let’s take a look at integration by parts for definite integrals The integration by parts formula for definite integrals is, Integration by Parts, Definite Integrals ∫ b a b u dv = uv a −... integration by parts on the first integral While that is a perfectly acceptable way of doing the problem it’s more work than we really need © 2007 Paul Dawkins 5 http://tutorial.math.lamar.edu/terms.aspx Calculus II to do Instead of splitting the integral up let’s instead use the following choices for u and dv ⎛t⎞ dv = cos ⎜ ⎟ dt ⎝4⎠ ⎛t⎞ v = 4sin ⎜ ⎟ ⎝4⎠ u = 3t + 5 du = 3 dt The integral is then, ⎛t⎞ ⎛t⎞ ⌠... the 5 integral are also multiplied by integration by parts problems © 2007 Paul Dawkins 1 5 Forgetting to do this is one of the more common mistakes with 6 http://tutorial.math.lamar.edu/terms.aspx Calculus II As this last example has shown us, we will sometimes need more than one application of integration by parts to completely evaluate an integral This is something that will happen so don’t get excited.. .Calculus II 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... use the method that you find easiest This may not be the method that others find easiest, but that doesn’t make it the wrong method © 2007 Paul Dawkins 8 http://tutorial.math.lamar.edu/terms.aspx Calculus II One of the more common mistakes with integration by parts is for people to get too locked into perceived patterns For instance, all of the previous examples used the basic pattern of taking u to... following u = x5 dv = x 3 + 1 dx However, as with the previous example this won’t work since we can’t easily compute v v = ∫ x 3 + 1 dx © 2007 Paul Dawkins 9 http://tutorial.math.lamar.edu/terms.aspx Calculus II This is not an easy integral to do However, notice that if we had an x2 in the integral along with the root we could very easily do the integral with a substitution Also notice that we do have... choices this time u = sin θ dv = eθ dθ du = cos θ dθ v = eθ The integral is now, ∫e θ © 2007 Paul Dawkins cos θ dθ = eθ cos θ + eθ sin θ − ∫ eθ cos θ dθ 10 http://tutorial.math.lamar.edu/terms.aspx Calculus II Now, at this point it looks like we’re just running in circles However, notice that we now have the same integral on both sides and on the right side it’s got a minus sign in front of it This means . covered in Calculus II however, we have since moved several of them into Calculus III. So, rather than split the chapter up I have kept it in the Calculus II notes and also put a copy in the Calculus. 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. 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