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 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


Slicing and Dicing Solids
Solids of Revolution 1: Disks
Solids of Revolution 2: Washers
Volumes Rotating About the $y$-axis

Integration by Parts

Behind IBP
Going in Circles
Tricks of the Trade

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
Completing the Square

Partial Fractions

Linear Factors
Quadratic Factors
Improper Rational Functions and Long Division

Strategies of Integration

Integration by Parts
Trig Integrals
Trig Substitutions
Partial Fractions

Improper Integrals

Type I Integrals
Type II Integrals
Comparison Tests for Convergence

Differential Equations

Separable Equations
Mixing and Dilution

Models of Growth

Exponential Growth and Decay
Logistic Growth

Infinite Sequences

Close is Good Enough (revisited)
Limit Laws for Sequences
Monotonic Convergence

Infinite Series

Geometric Series
Limit Laws for Series
Telescoping Sums and the FTC

Integral Test

Road Map
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

Alternating Series and the AS Test
Absolute Convergence

The Ratio and Root Tests

The Ratio Test
The Root Test

Strategies for testing Series

List of Major Convergence Tests

Power Series

Radius and Interval of Convergence
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

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
Order of Integration
Area and Volume Revisited


When talking about any mathematical quantity, it's important to keep track of three different questions:

  • What is it?
  • How do you compute it?
  • What is it good for?

This is especially true for integration, where the three questions have very different answers:

  • The integral $\int_a^b f(x) \,dx$ is the limit of a sum: $\displaystyle{\lim_{n\to\infty} \sum_{i=1}^n f(x_i^*)\,\Delta x}$.

  • We compute integrals with the Fundamental Theorem of Calculus: $\int_a^b f(x) \,dx = F(b)-F(a)$, where $F'(x)=f(x)$.

  • We use integration to compute the total amount of stuff in the interval $[a,b]$, where stuff could be area under a curve, distance traveled, volume of a solid of revolution, arc length of a curve, or a host of other applications.

Of these applications, the easiest to understand is area under a curve, which is why we used area to introduce integrals. Likewise, we will use volume to introduce double integrals. But remember: ordinary integrals are about much more than area, and double integrals are about much more than volume.

In the following video, we review the ideas and definitions of one-dimensional integrals. Not how to compute an integral, but what it is and what it's good for. On the next page, we'll see how the exact same ideas work in two and three dimensions. We'll tackle 'how do you compute it?' in the next learning module.

Remember the main idea of integration: The whole is the sum of the parts. Integration is a procedure for computing bulk quantities like area, volume, mass, distance, moment of inertia, and wealth. If $f(x)$ is the density of a quantity on interest, then

  1. Break the interval $[a,b]$ into $N$ pieces, each of size $\displaystyle\Delta x = \frac{b-a}{N}$. Label the pieces by an index $i$ that goes from 1 to $N$.

  2. For each $i$, pick a sample point $x_i^*$ somewhere in the $i$-th interval.

  3. Approximate the amount of stuff in the $i$-th piece as $f(x_i^*) \Delta x$, where $x_i^*$ is an arbitrary point in the $i$-th interval.

  4. Approximate the total amount of stuff as $\displaystyle{\sum_{i=1}^N f(x_i^*)\, \Delta x}$.

  5. To get an exact answer, take a limit as $N \to \infty$: $$\int_a^b f(x)\, dx = \lim_{N \to \infty} \,\sum_{i=1}^N f(x_i^*)\, \Delta x.$$

These ideas are illustated with the standard `area under a curve' example. The video on the right shows how this area can be approximated by adding up narrower and narrower rectangles, with $x_i^*$ being the left endpoint. (You may need to hit the 'replay' button a few times, since it's only 6 seconds long.)