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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

### Type I and Type II regions

If $R$ is a rectangle in the $x$-$y$ plane and $f(x,y)$ is a function defined on $R$ then we saw that $\displaystyle\iint_R f(x,y)\, dA$ is what we get when we

 Chop $R$ into a bunch of small boxes. Compute $f\left(x^*,y^*\right)\,\Delta x\, \Delta y$ for each box contained in $R$. Add up the boxes, and Take a limit as we chop $R$ into smaller and smaller pieces.

The exact same ideas apply when $R$ is an arbitrary region. As with rectangles, things get a lot simpler if we arrange the boxes intelligently.

In summary

 Type I regions are regions that are bounded by vertical lines $x=a$ and $x=b$, and curves $y=g(x)$ and $y=h(x)$, where we assume that $g(x) < h(x)$ and $a < b$. Then we can integrate first over $y$ and then over $x$:$$\iint_R f(x,y)\, dA = \int_{x=a}^b\, \int_{y=g(x)}^{h(x)} \,f(x,y)\, dy \, dx$$ Type II regions are bounded by horizontal lines $y=c$ and $y=d$, and curves $x=g(y)$ and $x=h(y)$, where we assume that $g(y)< h(y)$ and $c < d$. Then we can integrate first over $x$ and then over $y$:$$\iint_R f(x,y)\, dA = \int_{y=c}^d \,\int_{x=g(y)}^{h(y)} \,f(x,y)\, dx\, dy$$