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Calculus II - Notes Integration Techniques

Calculus II (Math 2414)

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 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.

 

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.

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.

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

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 - Notes Integration Techniques

Online Notes / Calculus II (Notes)

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