Home

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.