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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

### What is a double integral?

Integrals over rectangles are almost the same as integrals over intervals, except that now our density $f(x,y)$ is the amount of stuff per unit area instead of stuff per unit length.

The simplest example of "stuff" is volume, in which case $f(x,y)$ is height. So suppose that $R = [a,b] \times [c,d]$ is a rectangle, where $x$ runs from $a$ to $b$ and $y$ runs from $c$ to $d$. Let's figure out the volume of the solid between the $xy$ plane and the surface $z=f(x,y)$, and over the rectangle $R$.

 Break the interval $[a,b]$ into $m$ pieces, each of size $\displaystyle\Delta x = \frac{b-a}{m}$. Break the interval $[c,d]$ into $n$ pieces, each of size $\displaystyle\Delta y=\frac{d-c}{n}$. Together, this breaks $R$ into $nm$ smaller rectangular boxes, each of area $\Delta A =\Delta x\, \Delta y$. It breaks the solid into little towers of width $\Delta x$, depth $\Delta y$, and height $f(x,y)$. Label the boxes with two indices $i,j$, where $i$ says what column we're in and runs from 1 to $m$, while $j$ says what row we're in and runs from 1 to $n$. For each pair $i,j$, pick a sample point $\left(x_{ij}^*, y_{ij}^*\right)$ somewhere in the $ij$-th box. Approximate the volume of the tower over the $ij$-th box as $f\left(x_{ij}^*,y_{ij}^*\right) \,\Delta A$. Approximate the total volume as $$\displaystyle{\sum_{i=1}^m \sum_{j=1}^n f\left(x_{ij}^*, y_{ij}^*\right)\, \Delta A}.$$ To get an exact answer, take a limit as $n \to \infty$ and $m\to \infty$.

Interactive:

For each function, the integral of $f(x,y) dA$ is the volume of the region under the surface $z=f(x,y)$. The volume of each tower is $f\left(x_{ij}^*, y_{ij}^*\right) \Delta x \Delta y$. By varying the size of $\Delta x$ and $\Delta y$, and the choice of representative points $\left(x_{ij}^*, y_{ij}^*\right)$, you can see how the sums approximate the integral.

As with functions of one variable, the limit of this sum is the definition of a (double) integral.

 If $f(x,y)$ is a continuous function of two variables and if $R = [a,b] \times [c,d]$ is a rectangle, then the double integral of $f$ over $R$ is $$\iint_R f(x,y)\, dA = \iint_R f(x,y)\, dx \,dy = \lim_{m \to \infty}\,\lim_{n\to\infty} \,\sum_{i=1}^m \sum_{j=1}^n f\left(x_{ij}^*, y_{ij}^*\right)\, \Delta A.$$ A convenient choice for the sample point $\left(x_{ij}^*,y_{ij}^*\right)$ is given by the midpoint rule: $$x_{ij}^* = a + \left(i-\frac12\right)\,\Delta x, \qquad y_{ij}^* = c+\left(j-\frac12\right)\Delta y.$$

In the following video, we develop the same definition with a slightly different problem, namely computing the total population of the (conveniently rectangular) state of Colorado.