Calculus II Integral Calculus Lecture Notes Veselin Jungic Jamie Mulholland Department of Mathematics Simon Fraser University c Draft date January 2,
- If you use this textbook as a bibliographic reference, please include https://openstax org/details/books/calculus-volume-2 in your citation For questions
22 mai 2003 · fx(a, b) or fy(a, b) does not exist Definition 13 8 2 Let f be a function of two variables that has continuous second partial derivatives The
Now, I admit that Calculus II is harder than Calculus I Also, I may as well tell you that many — but not all — math students find it to be harder than the
Things You Should Know Coming Into Calc II Algebraic Rules, Properties, Formulas, Ideas and Processes: 1) Rules and Properties of Exponents
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
22 nov 2002 · course MATH 214-2: Integral Calculus I may keep working on this 1 2 2 Indefinite Integrals If F is an antiderivative of a function
Textbook: OpenStax Calculus Volume 2, 2016 edition, by Gilbert Strang, et al Page 2 Other Comments: The interplay between algebra and geometry should be
CALCULUS I, Second Semester VI Transcendental Functions 6 1 Inverse Functions The functions ex and lnx are inverses to each other in the sense that the
Calculus II SOCIAL SCIENCE PROGRAM COURSE OUTLINE FALL 2022 General Information Discipline: Mathematics Course code: 201-203-RE Ponderation: 3-2-3
........................................................................................................................................... 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
Integration Strategy ...................................................................................................................................52
Improper Integrals .....................................................................................................................................59
Comparison Test for Improper Integrals ...................................................................................................66
Approximating Definite Integrals .............................................................................................................73
Applications of Integrals ............................................................................................................. 80
Introduction ...............................................................................................................................................80
Arc Length ................................................................................................................................................81
..............................................................................................................................................87
Center of Mass ..........................................................................................................................................93
.................................................................................................................97
Probability ...............................................................................................................................................102
Parametric Equations and Polar Coordinates ........................................................................ 106
Introduction .............................................................................................................................................106
Parametric Equations and Curves ...........................................................................................................107
Tangents with Parametric Equations .......................................................................................................127
Area with Parametric Equations ..............................................................................................................134
Arc Length with Parametric Equations ...................................................................................................137
Surface Area with Parametric Equations.................................................................................................141
Polar Coordinates ....................................................................................................................................143
Tangents with Polar Coordinates ............................................................................................................153
Area with Polar Coordinates ...................................................................................................................155
Arc Length with Polar Coordinates .........................................................................................................162
Surface Area with Polar Coordinates ......................................................................................................164
Arc Length and Surface Area Revisited ..................................................................................................165
Sequences and Series ................................................................................................................. 167
Introduction .............................................................................................................................................167
Sequences ................................................................................................................................................169
More on Sequences .................................................................................................................................179
Series - The Basics .................................................................................................................................185
Series - Convergence/Divergence ..........................................................................................................191
Series - Special Series ............................................................................................................................200
Integral Test ............................................................................................................................................208
Comparison Test / Limit Comparison Test .............................................................................................217
Alternating Series Test ............................................................................................................................226
Absolute Convergence ............................................................................................................................232
Ratio Test ................................................................................................................................................236
Root Test .................................................................................................................................................243
Strategy for Series ...................................................................................................................................246
Estimating the Value of a Series .............................................................................................................249
Power Series ............................................................................................................................................260
Power Series and Functions ....................................................................................................................268
Taylor Series ...........................................................................................................................................275
Applications of Series .............................................................................................................................285
Binomial Series .......................................................................................................................................290
Vectors ........................................................................................................................................ 292
Introduction .............................................................................................................................................292
Vectors - The Basics ...............................................................................................................................293
Vector Arithmetic ...................................................................................................................................297
Dot Product .............................................................................................................................................302
Cross Product ..........................................................................................................................................310
Three Dimensional Space.......................................................................................................... 316
Introduction .............................................................................................................................................316
The 3-D Coordinate System ....................................................................................................................318
Equations of Lines ..................................................................................................................................324
Equations of Planes .................................................................................................................................330
Quadric Surfaces .....................................................................................................................................333
Functions of Several Variables ...............................................................................................................339
Vector Functions .....................................................................................................................................346
Calculus with Vector Functions ..............................................................................................................355
Tangent, Normal and Binormal Vectors .................................................................................................358
Arc Length with Vector Functions ..........................................................................................................362
Curvature .................................................................................................................................................365
Velocity and Acceleration .......................................................................................................................367
Cylindrical Coordinates ..........................................................................................................................370
Spherical Coordinates .............................................................................................................................372
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 i n 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 semes ter 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 oftenAlso, 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 conservespace 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.Let's start off with this section with a couple of integrals that we should already be able to do to
get us started. First let's take a look at the following. xx dx c ee So, that was simple enough. Now, let's take a look at, 2 x x dx e To do this integral we'll use the following substitution. 2 1 2 2 u xdu xdx xdxdugoing to be doing these kinds of substitutions in our head. If you have to stop and write these out
with every problem you will find that it will take you significantly longer to do these problems. Now, let's look at the integral that we really want to do. 6x x dx eThere is no substitution that we can use on this integral that will allow us to do the integral. So,
at this point we don't have the knowled ge to do this integral.To do this integral we will need to use integration by parts so let's derive the integration by parts
formula. We'll start with the product rule. fgf gf g