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.
- 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.
- 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.
- 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.
- 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.