### 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.**

Upon successful completion of this unit, you will be able to:

- use sigma notation to represent sums;
- approximate areas by Riemann sums;
- translate an area under a curve into a definite integral
- evaluate definite integrals geometrically using graphs of functions;
- determine whether a given function is integrable;
- find antiderivatives by changing the variable and using tables;
- use the Fundamental Theorem of Calculus to evaluate definite integrals
- differentiate integrals;
- solve applied problems that involve generalized area, that is, distance, work, and so forth;
- find an area between two curves; and
- find the average (mean) value of a function.

### 5.1: Introduction to Integration

Read this section on pages 1-7 to learn about area. Work through practice problems 1-9. For solutions to these problems, see page 10.

### 5.1.1: Practice Problems

Work through the odd-numbered problems 1-15 on pages 8-9. Once you have completed the problem set, check your answers for the odd-numbered questions against here.

### 5.1.2: Review

Before moving on, you should be comfortable with each of these topics:

- Area; pages 1-4.
- Applications of Area like Distance and Total Accumulation; pages 5-7.

- Area; pages 1-4.

### 5.2: Sigma Notation and Riemann Sums

Read this section on pages 1-10 to learn about area. Work through practice problems 1-9. For solutions to these problems, see pages 15-16.

### 5.2.1: Practice Problems

Work through the odd-numbered problems 1-61 on pages 10-15. Once you have completed the problem set, check your answers for the odd-numbered questions here.

### 5.2.2: Review

Before moving on, you should be comfortable with each of these topics:

- Sigma Notation; pages 1-2.
- Sums of Areas of Rectangles; pages 3-4.
- Area under a Curve - Riemann Sums; pages 5-8.
- Two Special Riemann Sums - Lower and Upper Sums; pages 9-10.

- Sigma Notation; pages 1-2.

### 5.3: The Definite Integral

Read this section on pages 1-6 to learn about the definite integral and its applications. Work through practice problems 1-6. For solutions to these problems, see page 11.

### 5.3.1: Practice Problems

Work through the odd-numbered problems 1-29 on pages 6-10. Once you have completed the problem set, check your answers for the odd-numbered questions here.

### 5.3.2: Review

Before moving on, you should be comfortable with each of these topics:

- The Definition of the Definite Integral; pages 1-3.
- Definite Integrals of Negative Functions; pages 3-5.
- Units for the Definite Integral; pages 5-6.

- The Definition of the Definite Integral; pages 1-3.

### 5.4: Properties of the Definite Integral

Read this section on pages 1-8 to learn about properties of definite integrals and how functions can be defined using definite integrals. Work through practice problems 1-5. For solutions to these problems, see page 11.

### 5.4.1: Practice Problems

Work through the odd-numbered problems 1-51 on pages 9-11. Once you have completed the problem set, check your answers for the odd-numbered questions against here.

### 5.4.2: Review

Before moving on, you should be comfortable with each of these topics:

- Properties of the Definite Integral; pages 1-2.
- Properties of Definite Integrals of Combinations of Functions; pages 3-5.
- Functions Defined by Integrals; pages 5-6.
- Which Functions Are Integrable?; pages 6-7.
- A Nonintegrable Function; page 8.

- Properties of the Definite Integral; pages 1-2.

### 5.5: Areas, Integrals, and Antiderivatives

Read this section on pages 1-6 to learn about the relationship among areas, integrals, and antiderivatives. Work through practice problems 1-5. For solutions to these problems, see pages 10-11.

### 5.5.1: Practice Problems

Work through the odd-numbered problems 1-25 on pages 7-9. Once you have completed the problem set, check your answers for the odd-numbered questions here.

### 5.5.2: Review

Before moving on, you should be comfortable with each of these topics:

- Area Functions as an Antiderivative; pages 1-2.
- Using Antiderivatives to Evaluate Definite Integrals; pages 2-4.
- Integrals, Antiderivatives, and Applications; pages 4-6.

- Area Functions as an Antiderivative; pages 1-2.

### 5.6: The Fundamental Theorem of Calculus

Read this section on pages 1-9 to see the connection between derivatives and integrals. Work through practice problems 1-5. For solutions to these problems, see pages 14-15.

### 5.6.1: Practice Problems

Work through the odd-numbered problems 1-67 on pages 10-13. Once you have completed the problem set, check your answers for the odd-numbered questions here.

### 5.6.2: Review

Before moving on, you should be comfortable with each of these topics:

- Antiderivatives; pages 1-3.
- Evaluating Definite Integrals; pages 4-5.
- Steps for Calculus Application Problems; pages 6-8.
- Leibnitz's Rule for Differentiating Integrals; page 9.

- Antiderivatives; pages 1-3.

### Problem Set 8

Take this assessment to see how well you understood these concepts.

- This assessment
**does not count towards your grade**. It is just for practice! - You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
- You can take this assessment as many times as you want, whenever you want.

- This assessment

### 5.7: Finding Antiderivatives

Read this section on pages 1-9 to see how one can (sometimes) find an antiderivative. In particular, we will discuss the change of variable technique. Change of variable, also called substitution or u-substitution (for the most commonly-used variable), is a powerful technique that you will use time and again in integration. It allows you to simplify a complicated function to show how basic rules of integration apply to the function. Work through practice problems 1-4. For solutions to these problems, see pages 12-13.

Watch these videos on change of variable, also called substitution or u-substitution.

### 5.7.1: Practice Problems

Work through the odd-numbered problems 1-69 on pages 10-12. Once you have completed the problem set, check your answers for the odd-numbered questions here.

### 5.7.2: Review

Before moving on, you should be comfortable with each of these topics:

- Indefinite Integrals and Antiderivatives; page 1.
- Properties of Antiderivatives (Indefinite Integrals); pages 2-3.
- Antiderivatives of More Complicated Functions; pages 3-4.
- Getting the Constant Right; pages 4-5.
- Making Patterns More Obvious - Changing Variables; pages 5-8.
- Changing the Variables and Definite Integrals; pages 8-9.
- Special Transformations - Antiderivatives of and ; page 9.

- Indefinite Integrals and Antiderivatives; page 1.

### 5.8: First Application of Definite Integral

Read this section on pages 1-8 to see how some applied problems can be reformulated as integration problems. Work through practice problems 1-4. For solutions to these problems, see pages 10-11.

### 5.8.1: Practice Problems

Work through the odd-numbered problems 1-41 on pages 8-10. Once you have completed the problem set, check your answers for the odd-numbered questions here.

### 5.8.2: Review

Before moving on, you should be comfortable with each of these topics:

- Area between Graphs of Two Functions; pages 1-4.
- Average (Mean) Value of a Function; pages 4-6.
- A Definite Integral Application - Work; pages 6-8.

- Area between Graphs of Two Functions; pages 1-4.

### 5.9: Using Tables to Find Antiderivatives

Read this section on pages 1-3 to learn how to use tables to find antiderivatives. See the Calculus Reference Facts for the table of integrals mentioned in the reading. Work through practice problems 1-5. For solutions to these problems, see page 6.

### 5.9.1: Practice Problems

Work through the odd-numbered problems 1-55 on pages 4-5. Once you have completed the problem set, check your answers for the odd-numbered questions here.

### 5.9.2: Review

Before moving on, you should be comfortable with each of these topics:

- Table of Integrals; pages 1-3.
- Using Recursive Formulas; page 3.

- Table of Integrals; pages 1-3.

### Problem Set 9

Take this assessment to see how well you understood these concepts.

- This assessment
**does not count towards your grade**. It is just for practice! - You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
- You can take this assessment as many times as you want, whenever you want.

- This assessment

### Appendix

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.