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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 Integrals

An improper integral of Type I is an integral whose limits of integration include $\infty$ or $-\infty$, or both. Remember that $\infty$ is a process (keep going and never stop), not a number! With this in mind, we define

 $$\int_a^\infty f(x) \,dx= \lim_{t \to \infty} \int_a^t f(x)\, dx,$$and likewise$$\int_{-\infty}^b f(x)\, dx = \lim_{t \to -\infty} \int_t^b f(x)\, dx.$$

When an integral runs from $-\infty$ to $\infty$, we have to break the integral into two pieces:$$\int_{-\infty}^\infty f(x)\, dx = \int_{-\infty}^a f(x) \,dx + \int_a^\infty f(x) \,dx,$$where we can choose any number for the break point $a$. (Zero is often convenient.) To evaluate the limits as $t \to \infty$ or $t \to -\infty$, we might need to use L'Hospital's rule.

#### On Convergence

 We'll say that $\displaystyle\int_a^\infty f(x)\,dx$ converges if $\displaystyle\lim_{t\to\infty}\int_a^t f(x)\,dx$ exists, and that the improper integral diverges if the limit doesn't exist. We define convergence/diverge of $\displaystyle\int_{-\infty}^b f(x)\,dx$ similarly in terms of the existence of its corresponding limit. For $\displaystyle\int_{-\infty}^\infty f(x)\,dx$ to converge we need both $\displaystyle\lim_{t\to\infty}\int_{a}^t f(x)\,dx$ and $\displaystyle\lim_{t\to-\infty}\int_{t}^a f(x)\,dx$ to exist.

We'll be talking a lot more about convergence and divergence when we get to sequences and series.

The following video explains Type I improper integrals and works out a number of examples.