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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

### Fubini's Theorem

The reasoning of the previous page can be summarized in a famous theorem:

 Fubini's Theorem: If $f(x,y)$ is a continuous function on a rectangle $R = [a,b] \times [c,d]$, then the double integral $\iint_R f(x,y)\, dA$ is equal to the iterated integral $$\int_c^d \left( \int_a^b f(x,y) \,dx\right )\, dy$$ and also to the iterated integral $$\int_a^b \left(\int_c^d f(x,y)\,dy \right )\, dx.$$ In both cases we can evaluate the inner integral with the fundamental theorem of calculus, treating the other variable as a constant.

Surprisingly enough, examples of unbounded and discontinuous functions $f(x,\,y)$ can be constructed where the iterated integral in one order is different from the iterated integral in the opposite order! But for any reasonable function, you can integrate in either order.

 Example: Find Compute $$\iint_R xe^y \,dA,$$ where $R$ is the rectangle $[0,2] \times [0,1]$. Solution 1: We can integrate first over $x$. \begin{eqnarray*} \iint_R xe^y \, dA & = & \int_0^1 \left (\int_0^2 xe^y\, dx \right )\, dy \cr \cr &=& \int_0^1 \left . \frac{x^2}{2} e^y \right |_{x=0}^2 \, dy \cr \cr & = & \int_0^1 2 e^y\, dy \ = \ 2e-2 \end{eqnarray*} Solution 2: We can integrate first over $y$. \begin{eqnarray*} \iint_R xe^y \, dA & = & \int_0^2 \left (\int_0^1 xe^y \,dy \right )\, dx \cr \cr &=& \int_0^2 \left . x e^y \right|_{y=0}^1 \, dx \cr \cr & = & \int_0^2 (e-1)x\, dx \cr & = & \left . \frac{(e-1)x^2}{2} \right |_0^2 = 2e-2. \end{eqnarray*}