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
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
Surface Area..............................................................................................................................................87
Center of Mass ..........................................................................................................................................93
Hydrostatic Pressure and Force .................................................................................................................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
© 2007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx
, Calculus II
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
© 2007 Paul Dawkins ii http://tutorial.math.lamar.edu/terms.aspx
, 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 http://tutorial.math.lamar.edu/terms.aspx
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
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
Surface Area..............................................................................................................................................87
Center of Mass ..........................................................................................................................................93
Hydrostatic Pressure and Force .................................................................................................................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
© 2007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx
, Calculus II
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
© 2007 Paul Dawkins ii http://tutorial.math.lamar.edu/terms.aspx
, 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 http://tutorial.math.lamar.edu/terms.aspx
