Home
Integration by Parts
Integration by Parts
Examples
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
Examples
Completing the Square
Partial Fractions
Introduction to Partial Fractions
Linear Factors
Irreducible Quadratic Factors
Improper Rational Functions and Long Division
Summary
Strategies of Integration
Substitution
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
Introduction
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
Ellipses
Hyperbolas
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
Introduction
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
Rearrangements
The Ratio and Root Tests
The Ratio Test
The Root Test
Examples
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
Differentiability
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 14
Examples 57
Swapping the Order of Integration
Area and Volume Revisited
Double integrals in polar coordinates
dA = r dr (d theta)
Examples
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
Jacobians
Examples
Bonus: Cylindrical and spherical coordinates


To determine whether a parametrized curve is concave up or concave
down, we need to see whether $d^2 y/dx^2$ is positive or negative. To
compute these higher derivatives, we start with the formula for
$dy/dx$ and apply the chain rule:
$$\frac{d\hbox{(anything)}}{dt} =
\frac{d\hbox{(anything)}}{dx}\frac{dx}{dt}.$$Dividing both sides by
$dx/dt$ gives$$\frac{d\hbox{(anything)}}{dx} = \frac{1}{dx/dt}
\frac{d\hbox{(anything)}}{dt}.$$
Since we are taking derivatives of many things with respect to $t$,
it is useful to have a special notation for time derivatives.
We use dots to denote time derivatives, much as
we use the prime symbol $'$ to mean a derivative with respect to $x$.
(Both the prime and dot notations are due to Isaac Newton.) That is,
$\dot x$ means $dx/dt$, $\ddot x$ means $d^2 x/dt^2$, and so on.

In
this notation, the chain rule is
$\displaystyle{\frac{d\hbox{(anything)}}{dx}
= \frac{\dot{\hbox{anything}}}{\dot x}}$, and in particular
$\displaystyle{\frac{dy}{dx} = \frac{\dot y}{\dot x}.}$ Taking a derivative
with respect to $x$ one more time and applying the quotient rule gives
\begin{eqnarray*}\frac{d^2y}{dx^2} = \frac{d}{dx}\frac{dy}{dx} &=& \frac{1}{\dot x} \frac{d (\dot y/\dot x)}{dt} \cr
&=& \frac{1}{\dot x} \frac{\dot x \ddot y  \dot y \ddot x}{\dot x^2} \cr
&=& \frac{\dot x \ddot y  \dot y \ddot x}{\dot x^3}.
\end{eqnarray*}
We can apply the same reasoning to get even higher derivatives, like
$$\frac{d^3y}{dx^3} = \frac{d}{dx}\frac{d^2y}{dx^2} =
\frac{1}{\dot x} \frac{d (d^2y/dx^2)}{dt},$$
but these come up a lot less often than the second derivative.
Example: Consider the curve $x=t^2$; $y=e^t$. For what values
of $t$ is $dy/dx$ positive? For what values of $t$ is $d^2y/dx^2$ positive?
Solution: We compute $\dot x = 2t$, $\ddot x=2$, $\dot y=e^t$,
$\ddot y=e^t$. Then $dy/dx = \dot y/\dot x = e^t/2t$. This is positive when
$t>0$ and negative when $t<0$ (and is not defined when $t=0$, at which time
the curve has a vertical tangent).
Meanwhile, $d^2 y/dx^2 = (\dot x \ddot y  \dot y \ddot x)/\dot x^3
= e^y(2t2)/8t^3$. This is positive when $t>1$ or $t<0$, is negative when
$0< t <1$, is zero when $t=1$, and is not defined when $t=0$.

