# Calculus II

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About *Calculus II:*

Here is a listing of all the material that is currently available online.

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 an how they can be used to simplify certain integrals.

Partial Fractions We will use partial fractions to allow us to do integrals involving 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 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.

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. We will also look at 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.

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.

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.

Curvature We will determine the curvature of a function in this section.

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.

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 an how they can be used to simplify certain integrals.

Partial Fractions We will use partial fractions to allow us to do integrals involving 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 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.

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. We will also look at 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.

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.

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.

Curvature We will determine the curvature of a function in this section.

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.