• 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 developing the intuition behind the notion of a tangent line. Integral calculus begins with understanding the intuition behind the idea of an area. We will be able to extend the notion of the area and apply these more general areas to various 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 10 hours.

• 5.1: Introduction to Integration

In this section, you will learn about the concept of Integration. Integral calculus applies to a wide range of applied problems, such as area.

• 5.2: Sigma Notation and Riemann Sums

Earlier, you learned about the concept of area. In this section, you will learn how to use notation to add a large set of values.

• 5.3: The Definite Integral

In this section, you will explore the Riemann Sums further, which will lead us to the concept of the definite integral. You will then learn about their applications.

• 5.4: Properties of the Definite Integral

In this section, you will continue to explore the definite integral by learning about the properties of integrals that sometimes define other functions.

• 5.5: Areas, Integrals, and Antiderivatives

In this section, you will explore the connection between areas, integrals, and antiderivatives, and how those connections can be used.

• 5.6: The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is one that you will use very often in calculating integrals.

• 5.7: Finding Antiderivatives

In this section, you will apply previous general knowledge of antiderivatives to find antiderivatives of complicated functions.

• 5.8: First Application of Definite Integral

Using your knowledge of definite integrals, we will now take a look at how to solve applied problems.

• 5.9: Using Tables to Find Antiderivatives

In this section, you will learn how to use the tables of integrals. You will find the table of integrals useful in learning calculus. Think about the table of integrals as a guide to help you solve problems.