About Course - Calculus II: Integral Calculus

   Video Introduction
   (Requires QuickTime)

You have just completed the first term of calculus, which dealt with the main idea of differentiation. As described, differentiation is the algebraic equivalent of taking a digital camera picture of a moving system to capture motion in an instant. This tool allowed Newton and Leibniz to calculate instantaneous rates of change within a dynamic system.

In this class, you will learn about the second major branch of calculus which integrates all of those instantaneous rates of change and creates the original function. The area located underneath the graph of the function provides vital information about that function, and the tools and techniques presented in this class will allow us to explore how to find that area easily and efficiently.

This course is designed to follow the College Board's Advanced Placement Calculus outline. Calculus II: Integral Calculus will build upon the foundation laid in Calculus I: Differential Calculus. If you have already taken a differential calculus course, you will find that Calculus II: Integral Calculus will have the same notation and vocabulary introduced in your previous course. After successful completion of this course, you will be prepared to take the advanced placement test for AB calculus which is offered by the College Board each spring, provided that you have followed the suggested timeline in the course outline for both calculus courses.

Introduction to Calculus

"Exactly what is calculus?" Calculus acts as a mathematical version of the Polaroid or digital camera. With a few neat tricks and techniques, virtually any moving object can be mathematically "stopped" at any given time and its motion analyzed by using calculus. Let's consider some history for a few minutes and see how calculus developed, beginning with Galileo.

With your science background, you know that Galileo Galilei was one of the premier scientists from the European Renaissance. Whether or not Galileo actually dropped objects from the Leaning Tower of Pisa in Italy, the legend that he did still persists. There is no doubt, though, that Galileo's theories of falling objects have stood the test of time.

Galileo proposed the idea of imagining vacuums when thinking through experiments. He also developed the "Scientific Method," which is the application, to experiments, of a logical progression of thought. His observations of motion provided an incredible impetus for the scientific minds that followed. Sir Isaac Newton is one of the most brilliant scientists who developed major scientific discoveries, based, in part, on Galileo's theories.

Among other things, Newton was interested in making mathematical measurements to support Galileo's empirical observations. In order to do so, however, Newton needed a mathematical tool that would operate in much the same way that a Polaroid or digital camera operates to stop motion in an instant. Imagine an object falling from a table to the floor, and think about what would happen if you could stop the falling motion at any given time. With the mathematical tool Newton sought, there would be a way to make some meaningful statements about how fast the object fell, how far the object fell, and in what amount of time. Calculus is the tool Newton developed to address his mathematical problem of how to analyze motion. For this reason, calculus is almost always associated with physics, and the languages used to describe both disciplines usually are intermingled.

Physicists have known for hundreds of years that calculus is the tool of choice to describe moving systems. Today, all sorts of motion are described, such as electron motion around the atomic nucleus, magnetic fields around a charge in a wire, and the orbits of planets about stars. Calculus allows us to describe systems of motion, and to examine how those systems are interdependent. Biologists examine population dynamics in ecosystems with complex mathematical and statistical models, and all relative movements within those systems are easily described using calculus. Economists examine the movements of money, prices, supply, and demand using mathematical models, and, you guessed it... calculus! Production managers use calculus tools to determine how much material should be used in fabrication assemblies to maximize given parameters while minimizing costs.

With all these incredible applications, is it any wonder that the study of calculus is a minimal foundation to virtually all science and mathematics pursuits? Many careers in the field of engineering require much more sophisticated mathematics, but all build on the solid foundation of calculus.

It is my hope that, as you pursue Calculus II: Integral Calculus, you won't lose sight of the reason calculus was developed, how calculus is used, and what mathematics can do for you in your own academic and career pursuits. Let's explore these ideas together as a class, and keep our minds open to learning from each other!

Prerequisites

A study of calculus requires some measure of self-discipline and self-motivation. One would assume that you, as a COOLSchool student, already possess these characteristics, but it bears repeating. As you consider if online calculus is a reasonable option, think about the following questions:

Are you skilled with a graphing calculator? The advanced placement calculus examination requires the use of a graphing calculator, and this course will demand a certain level of proficiency. For Calculus II: Integral Calculus, you will need a graphing calculator that will find integrals. In my face-to-face advanced math and advanced placement calculus classes, different types of graphing calculators are used, such as the Texas Instruments 83-plus and higher, as well as Sharps and Casios.

Have you taken mathematics through trigonometry and/or pre-calculus? While not absolutely necessary to understand the theory behind how the derivative works, you will need a working knowledge of radian measure and the relative changes observed in sine, cosine, and tangent values around a unit circle. In addition, familiarity with the trigonometric identities, calculations using "ex" and natural logarithms, and the general characteristics of a function are necessary to comfortably approach Calculus II: Integral Calculus without having to review or learn new material.

You will need to be able to read a textbook independently. As simple as this sounds, those of you interested in pursuing this course will need to be able to acquire information in a format that some, frankly, are not comfortable with. Rarely do "typical" students spend much time laboring over a paragraph or an example, trying to tease out the origins of every variable and the meaning of every expression. Calculus II: Integral Calculus requires a reasonable level of self-discipline with respect to the text!

Are you curious about mathematics and its application? Sounds strange, but as I tell my face-to-face high school classes, calculus is to college-level pursuits in mathematics and science as algebra I and geometry are to vocational-trade pursuits. Those of you interested in pursuing careers in science, engineering, and even economics will take calculus as the lowest level of required mathematics. Whether or not you acquire those skills here in COOLSchool, or later in college, a working knowledge of calculus becomes a major "gate-keeper" to a multitude of science-related career pathways.

Can you articulate questions? One unique aspect of an online class requires that you and your COOLSchool classmates TALK to one another via e-mail. You will need to formulate your questions and responses in a reasonable manner, appropriate to the material and to the forum. After class begins, you will learn how to participate in threaded discussions, which will become another important venue for sharing ideas.