Dave4Math / Courses / Theory of Integrals

Theory of Integrals

Master the theory of integrals. In this course, you'll learn about the indefinite and definite integrals, finding exact areas, and the fundamental theorems of calculus. 


January 11, 2021

Created by Dave

This course begins Summer 2021.
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Isn’t math complicated enough, so do you really need a teacher whose elaborate teaching style makes you feel like you’re a lab rat? Introducing Dave’s online math course: Theory of Integrals. Dave has a fun, easy-to-understand teaching style that compels you to succeed.

Who This Course is For

  • Students in college who want to learn more mathematics.
  • Anyone interested in the theory of integrals.
  • Students who are enrolled in calculus 1 and want to improve their grade.
  • Anyone interested in getting prepared for calculus 2.
  • Anyone wanting to learn about indefinite and definite integrals, the fundamental theorems of calculus, and numerical integration.


There is no required textbook, though you will need an up-to-date web browser, paper, and pen. A hand-held scientific calculator of your choice is required.

What You’ll Learn in Theory of Integrals

  • Antiderivatives
  • The Indefinite Integral
  • Basic Rules of Integration
  • Differential Equations
  • Initial Value Problems
  • How the Method of Substitution Works
  • The Technique of Integration by Substitution
  • More Examples
  • An Intuitive Look at Area
  • Sigma Notation
  • Summation Formulas
  • The Area Problem
  • Area and Distance
  • Definition of the Definite Integral
  • Defining the Area of the Region Under the Graph of a Function
  • Geometric Interpretation of the Definite Integral
  • The Definite Integral and Displacement
  • Properties of the Definite Integral
  • More General Definition of the Definite Integral
  • The Mean Value Theorem for Definite Integrals
  • The Fundamental Theorem of Calculus, Part I
  • The Fundamental Theorem of Calculus, Part II
  • Evaluating Definite Integrals Using Substitution
  • The Definite Integral as a Measure of Net Change
  • Definite Integrals of Odd and Even Functions
  • Approximating Definite Integrals
  • The Trapezoidal Rule
  • The Error in the Trapezoidal Rule
  • Simpson’s Rule
  • The Error in Simpson’s Rule

Course Description

This exciting course begins with a detailed look into antiderivatives. In particular, we study the basic integration formulas and discuss differential equations and initial value problems. After that, we expand our methods to add integration by substitution. We formalize this method and demonstrate how to use it through several examples. We include integrals of trigonometric functions and integrals of inverse trigonometric functions.

Next, we begin to motivate the definite integral through a serious look at the concept of area. We start with an intuitive, classical look and then proceed by studying sigma notation, finite sums and their properties, and limits of sums. After that, we end this lesson with a formal look at the area of the region under the graph of a function.

Finally, one of the main topics of calculus is reached: the Definite Integral. We begin with a formal definition of the definite integral and connect it to our previous examination of area. We illustrate the Riemann integral by using several examples, including a geometric interpretation of the definite integral and displacement. Several properties of the definite integral are proven and demonstrated through examples.

Towards the end of the course, we discuss (at length) the Fundamental Theorems of Calculus (no spoilers here). Then at the end, we study numerical integration techniques such as the trapezoidal rule and Simpson’s rule. The theorems are illustrated with many examples, and the errors involved with these methods are analyzed in detail.

Recommended Prerequisites for Theory of Integrals

I recommend the prerequisite course Applications of the Derivative. If you’re not sure if this course is for you, checkout the course contents below or find out more by checking out my free calculus 1 articles.

Course Content

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David A. Smith

Mathematics Educator

I sincerely believe that the potential in every student can be unlocked. As an accomplished and dedicated instructor, I have firm confidence that I can provide significant value to your studies.

I relish teaching and use it as a channel for my creativity as I seek to deliver real-life and relevant connections between the mathematics that I teach and my students. Promoting an active learning environment and excitement about mathematics is my dream come true.

David Smith (Dave) has a B.S. and M.S. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. David is the founder and CEO of Dave4Math.