Unit 5: The Integral
While previous units dealt with differential calculus, this unit starts the study of integral calculus. As you may recall, differential calculus began with the development of the intuition behind the notion of a tangent line. Integral calculus begins with understanding the intuition behind the notion of an area. In fact, we will be able to extend the notion of the area and apply these more general areas to a variety of problems. This will allow us to unify differential and integral calculus through the Fundamental Theorem of Calculus. Historically, this theorem marked the beginning of modern mathematics and is extremely important in all applications.
Completing this unit should take you approximately 32 hours.
5.1: Introduction to Integration
5.1.1: Practice Problems
5.2: Sigma Notation and Riemann Sums
5.2.1: Practice Problems
5.3: The Definite Integral
5.3.1: Practice Problems
5.4: Properties of the Definite Integral
5.4.1: Practice Problems
5.5: Areas, Integrals, and Antiderivatives
5.5.1: Practice Problems
5.6: The Fundamental Theorem of Calculus
5.6.1: Practice Problems
Problem Set 8
5.7: Finding Antiderivatives
5.7.1: Practice Problems
5.8: First Application of Definite Integral
5.8.1: Practice Problems
5.9: Using Tables to Find Antiderivatives
5.9.1: Practice Problems
Problem Set 9
By reviewing this course, you will have an invaluable list of references to assist you in solving future calculus problems after this course has ended. It is a standard experience, when solving calculus problems on your own, to react to the new problem with the following: "We did not solve that kind of problem in the course." Ah, but we did, in that the new problem is often a combination, or composition, of two problem types that were covered.
The course could not cover all possible trigonometric functions you will encounter. If you encounter a need for the derivative of , it is sufficient to recall that and that sine and cosine were covered. You can eventually become so good at this that future calculus problems can almost seem to be little more than plugging into formulas.
Engineering students who have to take several courses that involve the use of calculus are noted for having a Table of Integrals on their hip wherever they go, such as this one posted on Wikipedia.