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#### The Fundamental Theorem of Calculus

Three Different Quantities
The Whole as Sum of Partial Changes
The Indefinite Integral as Antiderivative
The FTC and the Chain Rule

#### The Indefinite Integral and the Net Change

Indefinite Integrals and Anti-derivatives
A Table of Common Anti-derivatives
The Net Change Theorem
The NCT and Public Policy

#### Substitution

Substitution for Indefinite Integrals
Revised Table of Integrals
Substitution for Definite Integrals

#### Area Between Curves

The Slice and Dice Principle
To Compute a Bulk Quantity
The Area Between Two Curves
Horizontal Slicing
Summary

#### Volumes

Slicing and Dicing Solids
Solids of Revolution 1: Disks
Solids of Revolution 2: Washers
Volumes Rotating About the $y$-axis

Behind IBP
Examples
Going in Circles

#### Integrals of Trig Functions

Basic Trig Functions
Product of Sines and Cosines (1)
Product of Sines and Cosines (2)
Product of Secants and Tangents
Other Cases

#### Trig Substitutions

How it works
Examples
Completing the Square

#### Partial Fractions

Introduction
Linear Factors
Improper Rational Functions and Long Division
Summary

#### Strategies of Integration

Substitution
Integration by Parts
Trig Integrals
Trig Substitutions
Partial Fractions

#### Improper Integrals

Type I Integrals
Type II Integrals
Comparison Tests for Convergence

#### Differential Equations

Introduction
Separable Equations
Mixing and Dilution

#### Models of Growth

Exponential Growth and Decay
Logistic Growth

#### Infinite Sequences

Close is Good Enough (revisited)
Examples
Limit Laws for Sequences
Monotonic Convergence

#### Infinite Series

Introduction
Geometric Series
Limit Laws for Series
Telescoping Sums and the FTC

#### Integral Test

The Integral Test
When the Integral Diverges
When the Integral Converges

#### 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 Test
The Root Test
Examples

#### Strategies for testing Series

List of Major Convergence Tests
Examples

#### Power Series

Finding the Interval of Convergence
Other Power Series

#### 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

What are Taylor Polynomials?
How Accurate are Taylor Polynomials?
What can go Wrong?
Other Uses of Taylor Polynomials

#### Partial Derivatives

Definitions and Rules
The Geometry of Partial Derivatives
Higher Order Derivatives
Differentials and Taylor Expansions

#### Multiple Integrals

Background
What is a Double Integral?
Volumes as Double Integrals

#### Iterated Integrals over Rectangles

One Variable at the Time
Fubini's Theorem
Notation and Order

#### Double Integrals over General Regions

Type I and Type II regions
Examples
Order of Integration
Area and Volume Revisited

### Notation and Order

Let's review our notation for the different kinds of integrals:

 $\displaystyle{\iint_R f(x,y) \, dA}$ is the double integral of $f(x,y)$ over the region $R$. $\displaystyle{\iint_R f(x,y) \, dx\, dy}$ and $\displaystyle{\iint_R f(x,y) \, dy \, dx}$ mean the exact same thing as $\displaystyle{\iint_R f(x,y) \, dA}$. The area of a little box is $\Delta A = \Delta x \,\Delta y = \Delta y\, \Delta x$, so the infinitesimal area is $dA = dx\, dy = dy\, dx$. $\displaystyle{\int_a^b \int_c^d f(x,y)\, dy \, dx}$ is an iterated integral. We first treat $x$ as a constant and integrate $f(x,y) \,dy$ from $y=c$ to $y=d$. Call the result $g(x)$. It is a function of $x$ and describes how much stuff is in the column with that value of $x$. We then integrate $g(x)\, dx$ from $x=a$ to $x=b$. $\displaystyle{\int_c^d \int_a^b f(x,y)\, dx \, dy}$ is an iterated integral where we first integrate over $x$ to get the contribution of a row, and then integrate over $y$ to add up all the rows. In an iterated integral, the order of $dx$ and $dy$ tells you which variable to integrate first. When in doubt, draw parentheses: $$\int_a^b \int_c^d f(x,y) \, dy\, dx = \int_a^b \left ( \int_c^d f(x,y) \,dy \right )\, dx.$$ Fubini's Theorem says that you can evaluate double integrals by doing an iterated integral in either order, but sometimes one order is a lot simpler than the other.

In the following video, we go over notation (2 minutes) and work two example problems (4 minutes each). The first example involves integrating $6x^2y$ over the rectangle $[1,3] \times [0,2]$, and both orders of integration are equally easy. The second example involves integrating $x e^{xy}$ over the same rectangle, and one order turns out to be much easier than the other.