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The Fundamental Theorem of Calculus

Three Different Concepts
The Fundamental Theorem of Calculus (Part 2)
The Fundamental Theorem of Calculus (Part 1)
More FTC 1

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
Examples to Try
Revised Table of Integrals
Substitution for Definite Integrals

Area Between Curves

Computation Using Integration
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
More Practice

Integration by Parts

Integration by Parts
Integration by Parts with a definite integral
Going in Circles
Tricks of the Trade

Integrals of Trig Functions

Antiderivatives of Basic Trigonometric Functions
Product of Sines and Cosines (mixed even and odd powers or only odd powers)
Product of Sines and Cosines (only even powers)
Product of Secants and Tangents
Other Cases

Trig Substitutions

How Trig Substitution Works
Summary of trig substitution options
Completing the Square

Partial Fractions

Linear Factors
Irreducible Quadratic Factors
Improper Rational Functions and Long Division

Strategies of Integration

Integration by Parts
Trig Integrals
Trig Substitutions
Partial Fractions

Improper Integrals

Type 1 - Improper Integrals with Infinite Intervals of Integration
Type 2 - Improper Integrals with Discontinuous Integrands
Comparison Tests for Convergence

Differential Equations

Separable Equations
Mixing and Dilution

Models of Growth

Exponential Growth and Decay
Logistic Growth

Infinite Sequences

Approximate Versus Exact Answers
Examples of Infinite Sequences
Limit Laws for Sequences
Theorems for and Examples of Computing Limits of Sequences
Monotonic Covergence

Infinite Series

Geometric Series
Limit Laws for Series
Test for Divergence and Other Theorems
Telescoping Sums and the FTC

Integral Test

Road Map
The Integral Test
Estimates of Value of the Series

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

The Ratio and Root Tests

The Ratio Test
The Root Test

Strategies for testing Series

Strategy to Test Series and a Review of Tests
Examples, Part 1
Examples, Part 2

Power Series

Radius and Interval of Convergence
Finding the Interval of Convergence
Power Series Centered at $x=a$

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

Taylor Polynomials
When Functions Are Equal to Their Taylor Series
When a Function Does Not Equal Its Taylor Series
Other Uses of Taylor Polynomials

Partial Derivatives

Visualizing Functions in 3 Dimensions
Definitions and Examples
An Example from DNA
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

How To Compute Iterated Integrals
Examples of Iterated Integrals
Cavalieri's Principle
Fubini's Theorem
Summary and an Important Example

Double Integrals over General Regions

Type I and Type II regions
Examples 1-4
Examples 5-7
Order of Integration


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.

The application of integration that is 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 that 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.

Some applications other than area and volume

Integration is a procedure for computing bulk quantities like area, volume, mass, distance, moment of inertia, and wealth. If $f(x)$ is a quantity, 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 $N \to \infty$ (here, it doesn't matter which $x$-value we choose in each interval): $$\int_a^b f(x)\, dx = \lim_{N \to \infty} \,\sum_{i=1}^N f(x_i)\, \Delta x.$$

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.)