Integration by Parts

Integration by Parts
Integration by Parts with a definite integral
Going in Circles
Tricks of the Trade

Integrals of Trig Functions

Antiderivatives of Basic Trigonometric Functions
Product of Sines and Cosines (mixed even and odd powers or only odd powers)
Product of Sines and Cosines (only even powers)
Product of Secants and Tangents
Other Cases

Trig Substitutions

How Trig Substitution Works
Summary of trig substitution options
Completing the Square

Partial Fractions

Introduction to Partial Fractions
Linear Factors
Irreducible Quadratic Factors
Improper Rational Functions and Long Division

Strategies of Integration

Integration by Parts
Trig Integrals
Trig Substitutions
Partial Fractions

Improper Integrals

Type 1 - Improper Integrals with Infinite Intervals of Integration
Type 2 - Improper Integrals with Discontinuous Integrands
Comparison Tests for Convergence

Modeling with Differential Equations

Separable Equations
A Second Order Problem

Euler's Method and Direction Fields

Euler's Method (follow your nose)
Direction Fields
Euler's method revisited

Separable Equations

The Simplest Differential Equations
Separable differential equations
Mixing and Dilution

Models of Growth

Exponential Growth and Decay
The Zombie Apocalypse (Logistic Growth)

Linear Equations

Linear ODEs: Working an Example
The Solution in General
Saving for Retirement

Parametrized Curves

Three kinds of functions, three kinds of curves
The Cycloid
Visualizing Parametrized Curves
Tracing Circles and Ellipses
Lissajous Figures

Calculus with Parametrized Curves

Video: Slope and Area
Video: Arclength and Surface Area
Summary and Simplifications
Higher Derivatives

Polar Coordinates

Definitions of Polar Coordinates
Graphing polar functions
Video: Computing Slopes of Tangent Lines

Areas and Lengths of Polar Curves

Area Inside a Polar Curve
Area Between Polar Curves
Arc Length of Polar Curves

Conic sections

Slicing a Cone
Parabolas and Directrices
Shifting the Center by Completing the Square

Conic Sections in Polar Coordinates

Foci and Directrices
Visualizing Eccentricity
Astronomy and Equations in Polar Coordinates

Infinite Sequences

Approximate Versus Exact Answers
Examples of Infinite Sequences
Limit Laws for Sequences
Theorems for and Examples of Computing Limits of Sequences
Monotonic Covergence

Infinite Series

Geometric Series
Limit Laws for Series
Test for Divergence and Other Theorems
Telescoping Sums

Integral Test

Preview of Coming Attractions
The Integral Test
Estimates for the Value of the Series

Comparison Tests

The Basic Comparison Test
The Limit Comparison Test

Convergence of Series with Negative Terms

Introduction, Alternating Series,and the AS Test
Absolute Convergence

The Ratio and Root Tests

The Ratio Test
The Root Test

Strategies for testing Series

Strategy to Test Series and a Review of Tests
Examples, Part 1
Examples, Part 2

Power Series

Radius and Interval of Convergence
Finding the Interval of Convergence
Power Series Centered at $x=a$

Representing Functions as Power Series

Functions as Power Series
Derivatives and Integrals of Power Series
Applications and Examples

Taylor and Maclaurin Series

The Formula for Taylor Series
Taylor Series for Common Functions
Adding, Multiplying, and Dividing Power Series
Miscellaneous Useful Facts

Applications of Taylor Polynomials

Taylor Polynomials
When Functions Are Equal to Their Taylor Series
When a Function Does Not Equal Its Taylor Series
Other Uses of Taylor Polynomials

Functions of 2 and 3 variables

Functions of several variables
Limits and continuity

Partial Derivatives

One variable at a time (yet again)
Definitions and Examples
An Example from DNA
Geometry of partial derivatives
Higher Derivatives
Differentials and Taylor Expansions

Differentiability and the Chain Rule

The First Case of the Chain Rule
Chain Rule, General Case
Video: Worked problems

Multiple Integrals

General Setup and Review of 1D Integrals
What is a Double Integral?
Volumes as Double Integrals

Iterated Integrals over Rectangles

How To Compute Iterated Integrals
Examples of Iterated Integrals
Fubini's Theorem
Summary and an Important Example

Double Integrals over General Regions

Type I and Type II regions
Examples 1-4
Examples 5-7
Swapping the Order of Integration
Area and Volume Revisited

Double integrals in polar coordinates

dA = r dr (d theta)

Multiple integrals in physics

Double integrals in physics
Triple integrals in physics

Integrals in Probability and Statistics

Single integrals in probability
Double integrals in probability

Change of Variables

Review: Change of variables in 1 dimension
Mappings in 2 dimensions
Bonus: Cylindrical and spherical coordinates

Ellipses can be elegantly described in four ways.

  1. Via Cartesian (rectangular) coordinates.
  2. In terms of distances to two foci (plural of focus).
  3. In terms of distances to a focus and a directrix.
  4. In polar coordinates.

We will do the first two on this page, and the third and fourth later on.

The simplest description of an ellipse is as a squashed or stretched circle. Start with the unit circle $x^2 + y^2 =1$, and stretch it by a factor of $a$ in the $x$ direction and $b$ in the $y$ direction to get:

The standard formula for an ellipse in rectangular coordinates is $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.$$

The points $(\pm a,0)$ (and sometimes the points $(0,\pm b)$) are called vertices. If $a > b > 0$, then the major axis is the line segment from $(-a,0)$ to $(a,0)$ and the semi-major axis is the line segment from the origin to $(a,0)$. Likewise, the minor axis runs from $(0,-b)$ to $(0,b)$ and the semi-minor axis runs from the origin to $(0,b)$. If $b > a > 0$, then the major and semi-major axes are vertical and the minor and semi-minor axes are horizontal. For now we'll stick with the case that $a > b$, so that the ellipse is short and fat. The origin is the center of the ellipse.

Now let $c = \sqrt{a^2-b^2}$. The points $(\pm c, 0)$ are called foci. These points are extremely important in astronomy, since planets follow elliptical orbits with the sun at a focus, not with the sun at the center. Let $F_1(-c,0)$ and $F_2(c,0)$ be the two foci, let $P(x,y)$ be an arbitrary point on the ellipse. Let $L_1$ be the distance from $F_1$ to $P$, and let $L_2$ be the distance from $F_2$ to $P$, as in the figure on the right.

Amazing fact: The ellipse is the set of all points where $L_1 + L_2 = 2a$.

This fact gives elliptical rooms amazing acoustic properties. If you whisper at one focus of such a room, the sound waves from your voice will bounce off the walls and converge at the other focus -- that's why it is called a focus. The same goes for light reflecting off elliptical mirrors.

To understand the amazing fact, let's convert the equation $L_1 + L_2 = 2a$ to rectangular coordinates: \begin{eqnarray*} L_1 + L_2 & = & 2a \cr\cr L_1 & = & 2a-L_2 \cr \cr \sqrt{(x+c)^2+y^2} & = & 2a -\sqrt{(x-c)^2 + y^2} \cr\cr (x+c)^2 + y^2 & = & 4a^2 + (x-c)^2 + y^2 - 4a \sqrt{(x-c)^2 + y^2}\cr\cr 4a\sqrt{(x-c)^2 + y^2}&=& 4a^2-4cx \cr \cr a \sqrt{(x-c)^2 + y^2} &=& a^2-cx \cr \cr a^2(x-c)^2+ a^2 y^2 &=& a^4+c^2x^2 -2a^2cx \cr \cr a^2x^2 + a^2c^2 -2a^2cx + y^2 &=& a^4 + c^2x^2 -2a^2cx \cr \cr (a^2-c^2)x^2 + a^2 y^2 &=& a^2(a^2-c^2) \cr \cr b^2 x^2 + a^2 y^2 &=& a^2b^2 \cr \cr \frac{x^2}{a^2} + \frac{y^2}{b^2} &=& 1,\end{eqnarray*} where we have used the fact that $b^2=a^2-c^2$. That's a long and messy calculation for a simple and elegant result. You should be able to construct the equation of an ellipse given any two of $a$, $b$ and $c$, since you can get the third from $c^2=a^2-b^2.$

  Example 1: Find the location of the foci of the ellipse $\displaystyle{\frac{x^2}{25} + \frac{y^2}{9}=1}$.

Solution: We have $a=5$ and $b=3$, so $c = \sqrt{a^2-b^2} = 4$. The foci are at $(\pm 4,0)$.

  Example 2: Find the equation of an ellipse with foci at $(\pm 3,0)$ if $b=4$.

Solution: Since $c=3$ and $b=4$, $a^2=3^2+4^2=25$, so $a=5$. This makes the equation $$\frac{x^2}{25} + \frac{y^2}{16} = 1.$$

The ratio $c/a$ is called the eccentricity of the ellipse, and is usually denoted $e$. Note that $e < 1$. A circle can be viewed as an ellipse with eccentricity zero, and with both foci at the origin.

It is easy to plot an ellipse as a parametrized curve. Just take $$x = a \cos(t); \qquad y = b\sin(t),$$ with the parameter $t$ running from $0$ to $2\pi$.