This course is designed to introduce you to the study of calculus. You will learn concrete applications of how calculus is used and, more importantly, why it works. Calculus is not a new discipline; it has been around since the days of Archimedes. However, Isaac Newton and Gottfried Leibniz, two seventeenth-century European mathematicians concurrently working on the same intellectual discovery hundreds of miles apart, were responsible for developing the field as we know it today. This brings us to our first question, what is today's calculus? In its simplest terms, calculus is the study of functions, rates of change, and continuity. While you may have cultivated a basic understanding of functions in previous math courses, in this course you will come to a more advanced understanding of their complexity, learning to take a closer look at their behaviors and nuances. In this course, we will address three major topics: limits, derivatives, and integrals, as well as study their respective foundations and applications. By the end of this course, you will have a solid understanding of the behavior of functions and graphs. Whether you are entirely new to calculus or just looking for a refresher on a particular topic, this course has something to offer, balancing computational proficiency with conceptual depth.

## Topic outline

- Course Introduction
- Unit 1: Analytic Geometry
Most of the material in this unit will be review. However, the notions of points, lines, circles, distance, and functions will be central in everything that follows. Lines are basic geometric objects which will be of great importance in the study of differential calculus, particularly in the study of tangent lines and linear approximations.

We will also take a look at the practical uses of mathematical functions. This course will use mathematical models, or structures, that predict practical situations in order to describe and study a number of real-life problems and situations. They are essential to the development of every major business and every scientific field in the modern world.

**Completing this unit should take you approximately 9 hours.** - 1.1: Lines
Read Section 1.1 (pages 14-17). Working with lines should be familiar to you, and this section serves as a review of the notions of points, lines, slope, intercepts, and graphing.

Work through problems 1-18. When you are done, check your answers in Appendix A.

- 1.2: Distance between Two Points, Circles
Read Section 1.2 (pages 19 and 20). This reading reviews the notions of distance in the plane and the equations and graphs of circles.

Please click on the link above, and work through problems 1, 2, and 6 in Exercise 1.2. When you are done, check your answers in Appendix A.

Click the link to Book I, and click on the "Index" button. Click on button 5 (Distance) to launch the first module. Complete problems 1-5. Then, return to the index and click on button 8 (Circles II) to launch the other module. Complete problems 1-5. If at any time the problem set becomes too easy for you, feel free to move on.

- 1.3: Functions
Read Section 1.3 (pages 20-24). This reading reviews the notion of functions, linear functions, domain, range, and dependent and independent variables.

Work through problems 1-16 for Exercise 1.3. When you are done, check your answers in Appendix A.

- 1.4: Shifts and Dilations
Click the link to Book I, and then click on the "Index" button. Click on button 18 (Transforming Graphs) to launch the module. Complete problems 1-15. If at any time the problem set becomes too easy for you, feel free to move on.

Read Section 1.4 (pages 25-28). This reading will review the graph transformations of shifts and dilations associated to some basic ways of manipulating functions.

- Unit 2: Instantaneous Rate of Change: The Derivative
In this unit, you will study the instantaneous rate of change of a function. Motivated by this concept, you will develop the notion of limits, continuity, and the derivative. The limit asks the question, "What does the function do as the independent variable becomes closer and closer to a certain value?" In simpler terms, the limit is the

*natural tendency*of a function. The limit is incredibly important due to its relationship to the derivative, the integral, and countless other key mathematical concepts. A strong understanding of the limit is essential to success in the field of mathematics.A derivative is a description of how a function changes as its input varies. In the case of a straight line, this derivative, or slope, is the same at every point, which is why we can describe the slope of an entire function with one number when it is linear. You will learn that we can do the same for nonlinear functions. The slope, however, will not be constant; it will change as the independent variable changes.

**Completing this unit should take you approximately 16 hours.** - 2.1: The Slope of a Function
In this subunit, you will look at the first of two major problems that lie at the heart of calculus: the tangent line problem. This intellectual exercise demonstrates the origins of derivatives for nonlinear functions.

Read Section 2.1 (pages 29-33). You will be introduced to the notion of a derivative through studying a specific example. The example will also reveal the necessity of having a precise definition for the limit of a function.

Watch this video. In this lecture, Jerison introduces the notion of a derivative as the rate of change of a function, or the slope of the tangent line to a function at a point.

Work through problems 1-6 for Exercise 2.1. When you are done, check your answers against Appendix A.

- 2.2: An Example
Read Section 2.2 (pages 34-36). This reading discusses the derivative in the context of studying the velocity of a falling object. This example again motivates the need for a more rigorous approach to the concept of a limit.

Work through problems 1-3 for Exercise 2.2. When you are done, check your answers against Appendix A.

- 2.3: Limits
In this subunit, you will take a close look at a concept that you have used intuitively for several years: the limit. The limit asks the question, "what does the function do as the independent variable becomes closer and closer to a certain value?" In simpler terms, the limit is the natural tendency of a function. The limit is incredibly important due to its relationship to the derivative, the integral, and countless other key mathematical concepts. A strong understanding of the limit is essential to success in the field of mathematics.

- 2.3.1: The Definition and Properties of Limits
Read Section 2.3 (pages 36-45). Read this section carefully, and pay close attention to the definition of the limit and the examples that follow. You should also closely examine the algebraic properties of limits as you will need to take advantage of these in the exercises.

Watch this video.

Work through problems 1-18 for Exercise 2.3. When you are done, check your answers against Appendix A.

Work through problems 1-10. When you are done, check your solutions against the answers provided.

- 2.3.2: The Squeeze Theorem
Read section 4.3 (pages 75-77). The Squeeze Theorem is an important application of the limit concept and is useful in many limit computations. This reading teaches you a useful trick for calculating limits of functions where at first glance you might seem to be dividing by zero.

Watch this brief video. The creator of this video describes and illustrates the Squeeze Theorem by using specific examples.

- 2.4: The Derivative Function
Read Section 2.4 (pages 46-50). In this reading, you will see how limits are used to compute derivatives. A derivative is a description of how a function changes as its input varies. In the case of a straight line, this description is the same at every point, which is why we can describe the slope (another word for the derivative) of an entire function with one slope when it is linear. You will learn that we can do the same for nonlinear functions. The slope, however, will not be constant; it will change as the independent variable changes.

Work through problems 1-5 and 8-11 for Exercise 2.4. When you are done, check your answers against Appendix A.

- 2.5: Adjectives for Functions
Read Section 2.5 (pages 51-54). The intuitive notion of a continuous function is made precise using limits. In addition, you will be introduced to the Intermediate Value Theorem, which rigorously captures the intuitive behavior of continuous real-valued functions.

- 2.5.1: Continuous Functions
Click the link to Book I, and click on the "Index" button. Click on button 26 (A missing value). Work on problems 15-26. Return to the "Index," and click on button 27 (Discontinuities of simple piecewise defined functions). Complete problems 1-10. If at any time the problem set becomes too easy for you, feel free to move on.

- 2.5.2: Differentiable Functions
Click the link to Book I, and click on the "Index" button. Click on button 39 (Differentiability). Complete problems 1-10. If at any time the problem set becomes too easy for you, feel free to move on.

- 2.5.3: The Intermediate Value Theorem
Watch this brief video for an explanation on the Intermediate Value Theorem.

- Unit 3: Rules for Finding Derivates
Computing a derivative requires the computation of a limit. Because limit computations can be rather involved, we like to minimize the amount of work we have to do in practice. In this unit, we build up some rules for differentiation which will speed up our calculations of derivatives. In particular, you will see how to differentiate the sum, difference, product, quotient, and composition of two (or more) functions. You will also learn rules for differentiating power functions (including polynomial and root functions).

**Completing this unit should take you approximately 12 hours.** - 3.1: The Power Rule
Read Section 3.1 (pages 55-57). Here, you will learn a simple rule for finding the derivative of a power function without explicitly computing a limit.

Work through problems 1-6 in Exercise 3.1. When you are done, check your answers against Appendix A.

- 3.2: Linearity of the Derivative
Read Section 3.2 (pages 58 and 59). In this reading, you will see how the derivative behaves with regard to addition and subtraction of functions and with scalar multiplication. That is, you will see that the derivative is a linear operation.

Work through problems 1-9, 11, and 12 for Exercise 3.2. When you are done, check your answers against Appendix A.

- 3.3: The Product Rule
Read Section 3.3 (pages 60 and 61). The naïve assumption is that the derivative of a product of two functions is the product of the derivatives of the two functions. This assumption is false. In this reading, you will see that the derivative of a product is slightly more complicated, but that it follows a definite rule, called the product rule.

Work through problems 1-5 for Exercises 3.3. When you are done, check your answers against Appendix A.

Click the link to Book I, and click on the "Index" button. Click on button 44 (Product Rule). Complete problems 1-10. If at any time the problem set becomes too easy for you, feel free to move on.

- 3.4: The Quotient Rule
Read Section 3.4 (pages 62-65). As with products, the derivative of a quotient of two functions is not simply the quotient of the two derivatives. In this reading, you will see the quotient rule for differentiating a quotient of two functions. In particular, this will allow you to find the derivative of any rational function.

Work through problems 5, 6, 8, and 9 for Exercise 3.4. When you are done, check your answers against Appendix A.

Click the link to Book I, and click on the "Index" button. Click on button 45 (Quotient Rule) to launch the module. Complete problems 1-10. If at any time the problem set becomes too easy for you, feel free to move on.

- 3.5: The Chain Rule
Read Section 3.5 (pages 65-69). The chain rule explains how the derivative applies to the composition of functions. Pay particular attention to Example 3.11, as it works through a derivative computation where all of the differentiation rules of this unit are applied in finding the derivative of one function.

Watch this video.

Work through problems 1-20 and 36-39 for Exercise 3.5. When you are done, check your answers against Appendix A.

- Unit 4: Transcendental Functions
In this unit, you will investigate the derivatives of trigonometric, inverse trigonometric, exponential, and logarithmic functions. Along the way, you will develop a technique of differentiation called implicit differentiation. Aside from allowing you to compute derivatives of inverse function, implicit differentiation will also be important in studying related rates problems later on.

**Completing this unit should take you approximately 17 hours.** - 4.1: Trigonometric Functions
Read Section 4.1 (pages 71-74). This reading will review with you the definition of trigonometric functions.

Work through problems 1-4 and 11 for Exercise 4.1. When you are done, check your answers against Appendix A.

- 4.2: The Derivative of Sine
Read Section 4.2 (pages 74 and 75). This reading begins the computation of the derivative of the sine function. Two specific limits will need to be evaluated in order to complete the computation. These limits are addressed in the following section.

- 4.3: A Hard Limit
Read Section 4.3 (pages 75-77). You read this section before to become acquainted with the Squeeze Theorem. When you read the section again, pay particular attention to the geometric argument used to set up the application of the Squeeze Theorem.

Work through problems 1-7 for Exercise 4.3. When you are done, check your answers against Appendix A.

- 4.4: The Derivative of Sine, Continued
- Read Section 4.4 (pages 77 and 78). This reading completes the computation of the derivative of the sine function. Be sure to review all of the concepts which are involved in this computation.
Work through problems 1-5 for Exercise 4.4. When you are done, check your answers against Appendix A.

- 4.5: Derivatives of the Trigonometric Functions
Read Section 4.5 (pages 78 and 79). Building on the work done to compute the derivative of the sine function and the rules of differentiation from previous readings, the derivatives of the remaining trigonometric functions are computed.

Work through problems 1-18 for Exercise 4.5. When you are done, check your answers against Appendix A.

- 4.6: Exponential and Logarithmic Functions
Read Section 4.6 (pages 80 and 81). This reading reviews the exponential and logarithmic functions, their properties, and their graphs.

- 4.7: Derivatives of the Exponential and Logarithmic Functions
Read Section 4.7 (pages 82-86). In this reading, the derivatives of the exponential and logarithmic functions are computed. otice that along the way the number e is defined in terms of a particular limit.

Watch this video. In this video, Jerison makes use of implicit differentiation at times during this lecture.

Work through problems 1-15 and 20 for Exercise 4.7. When you are done, check your answers against Appendix A.

- 4.8: Implicit Differentiation
Read Section 4.8 (pages 87-90). As a result of the chain rule, we have a method for computing derivatives of curves which are not explicitly described by a function. This method, called implicit differentiation, allows us to find tangent lines to such curves.

Watch this video.

Work through problems 1-9 and 11-16 for Exercise 4.8. When you are done, check your answers against Appendix A.

- 4.9: Inverse Trigonometric Functions
Read Section 4.9 (pages 91-94). In this reading, implicit differentiation and the Pythagorean identity are used to compute the derivatives of inverse trigonometric functions. You should notice that the same techniques can be used to find derivatives of other inverse functions as well.

Work through problems 3-11 for Exercise 4.9. When you are done, check your answers against Appendix A.

- 4.10: Limits Revisited
Read Section 4.10 (pages 94-97). You will learn how derivatives relate back to limits. Limits of Indeterminate Forms (or limits of functions that, when evaluated, tend to 0/0 or ∞/∞) have previously been beyond our grasp. Using L'Hopital's Rule, you will find that these limits are attainable with derivatives.

Work through problems 1-10 and 21-24 for Exercise 4.10. When you are done, check your answers against Appendix A.

- 4.11: Hyperbolic Functions
Read Section 4.11 (pages 99-102). In this reading, you are introduced to the hyperbolic trigonometric functions. These functions, which appear in many engineering and physics applications, are specific combinations of exponential functions which have properties similar to those that the ordinary trigonometric functions have.

- Unit 5: Curve Sketching
This unit will ask you to apply a little critical thinking to the topics this course has covered thus far. To properly sketch a curve, you must analyze the function and its first and second derivatives in order to obtain information about how the function behaves, taking into account its intercepts, asymptotes (vertical and horizontal), maximum values, minimum values, points of inflection, and the respective intervals between each of the above. After collecting this information, you will need to piece it all together in order to sketch an approximation to the graph of the original function.

**Completing this unit should take you approximately 10 hours.** - 5.1: Maxima and Minima
Read Section 5.1 (pages 103-106). Fermat's Theorem indicates how derivatives can be used to find where a function attains its highest or lowest points.

Watch this video from 30:00 to the end. The lecture will make use of the first and second derivative tests, which you will read about below.

Work through problems 1-12 and 15 for Exercise 5.1. When you are done, check your answers against Appendix A.

- 5.2: The First Derivative Test
Read Section 5.2 (page 107). In this reading, you will see how to use information about the derivative of a function to find local maxima and minima.

Work through problems 1-15 for Exercise 5.2. When you are done, check your answers against Appendix A.

- 5.3: The Second Derivative Test
Read Section 5.3 (pages 108 and 109). In this reading, you will see how to use information about the second derivative (that is, the derivative of the derivative) of a function to find local maxima and minima.

Work through problems 1-10 for Exercise 5.3. When you are done, check your answers against Appendix A.

- 5.4: Concavity and Inflection Points
Read Section 5.4 (pages 109 and 110). In this reading, you will see how the second derivative relates to concavity of the graph of a function and use this information to find the points where the concavity changes, i.e. the inflection points of the graph.

Work through problems 1-9 and 19 for Exercise 5.4. When you are done, check your answers against Appendix A.

- 5.5: Asymptotes and Other Things to Look For
Read Section 5.5 (pages 111 and 112). In this reading, you will see how limits can be used to find any asymptotes the graph of a function may have.

Watch this lecture until 45:00. The majority of the video lecture is about curve sketching, despite the title of the video.

Work through problems 1-5 and 15-19 for Exercise 5.5. When you are done, graph the curves using Wolfram Alpha to check your answers.

- Unit 6: Applications of the Derivative
With a sufficient amount of sophisticated machinery under your belt, you will now start to look at how differentiation can be used to solve problems in various applied settings. Optimization is an important notion in fields like biology, economics, and physics when we want to know when growth is maximized, for example.

In addition to providing methods to solve problems directly, the derivative can also be used to find approximate solutions to problems. You will explore two such methods in this section: Newton's method and the method of differentials.

**Completing this unit should take you approximately 13 hours.** - 6.1: Optimization
Read Section 6.1 (pages 115-124). An important application of the derivative is to find where a function takes its global maximum and its global minimum. The Extreme Value Theorem indicates how to approach this problem. Pay particular attention to the summary at the end of the section.

Watch this video until 45:00. The majority of the lecture is about optimization, despite its title.

Work through problems 5, 7, 9, 10, 14, 16, 22, 26, 28, and 33 for Exercise 6.1. When you are done, check your answers against Appendix A.

- 6.2: Related Rates
You now know how to take the derivative with respect to the independent variable. In other words, you know how to determine a function's rate of change when given the input's rate of change. But what if the independent variable was itself a function? What if, for example, the input was a function of time? How do we identify how the function changes as time changes? This subunit will explore the answers to these questions.

Read Section 6.2 (pages 127-132). Another application of the chain rule, related rates problems apply to situations where multiple dependent variables are changing with respect to the same independent variable. Make note of the summary in the middle of page 128.

Watch this video until 40:30. The majority of the lecture is about related rates, despite its title.

Work through problems 1, 3, 5, 11, 14, 16, 19-21, and 25 for Exercise 6.2. When you are done, check your answers against Appendix A.

- 6.3: Newton's Method
Newton's Method is a process by which we estimate the roots of a real-valued function. You may remember the bisection method, where we find a root by creating smaller and smaller intervals. Newton's Method uses the derivative in order to account for both the speed at which the function changes and its actual position. This creates an algorithm that can help us identify the location of roots even more quickly.

Newton's Method requires that you start "sufficiently close" (a somewhat arbitrary specification that varies from problem to problem) to the actual root in order to estimate it with accuracy. If you start too far from the root, then the algorithm can be led awry in certain situations.

Read Section 6.3 (pages 135-138). In this reading, you will be introduced to a numerical approximation technique called Newton's Method. This method is useful for finding approximate solutions to equations which cannot be solved exactly.

Watch this video until 15:10. This portion of the video is about Newton's Method, despite the title.

Work through problems 1-4 for Exercise 6.3. When you are done, check your answers against Appendix A.

- 6.4: Linear Approximations
In this subunit, you will learn how to estimate future data points based on what you know about a previous data point and how it changed at that particular moment. This concept is extremely useful in the field of economics.

Read Section 6.4 (pages 139 and 140). In this reading, you will see how tangent lines can be used to locally approximate functions.

Watch this video until 39:00. Beyond 39:00, Jerison discusses quadratic approximations to functions, which are in a certain sense one step beyond linear approximations. If you are interested, please continue viewing the lecture to the end.

Work through problems 1-4 for Exercise 6.4. When you are done, check your answers against Appendix A.

- 6.5: The Mean Value Theorem
Read Section 6.5 (pages 141-144). The Mean Value Theorem is an important application of the derivative which most often is used in further developing mathematical theories. A special case of the Mean Value Theorem, called Rolle's Theorem, leads to a characterization of antiderivatives.

Watch this video from 15:10 to the end.

Work through problems 1, 2, and 6-9 for Exercise 6.5. When you are done, check your answers against Appendix A.

- Unit 7: Integration
In this unit of the course, you will learn about integral calculus, a subfield of calculus that studies the area formed under the curve of a function. Though not necessarily intuitive, this concept is closely related to the derivative, which you will revisit in this unit.

**Completing this unit should take you approximately 21 hours.** - 7.1: Motivation
Read Section 7.1 (pages 145-149). This reading motivates the integral through two examples. The first addresses the question of how to determine the distance traveled based only on information about velocity. The second addresses the question of how to determine the area under the graph of a function. Surprisingly, these two questions are closely related to each other and to the derivative.

Watch this video.

- Work through problems 1-8 for Exercise 7.1. When you are done, check your answers against Appendix A.

- 7.2: The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus is the apex of our course. It explains the relationship between the derivative and the integral, tying the two major facets of this course together. In the previous section, you learned the definition of the definite integral as a limit of a Riemann Sum. The computations were long and involved. In this subunit, you will learn about the Fundamental Theorem of Calculus, which makes computation of definite integrals significantly easier.

Read Section 7.2 (pages 149-155). Pay close attention to the treatment of Riemann sums, which lead to the definite integral. The Fundamental Theorem of Calculus explicitly describes the relationship between integrals and derivatives.

Watch these videos.

Work through problems 7-22 for Exercise 7.2. When you are done, check your answers against Appendix A.

- 7.3: Some Properties of Integrals
Read Section 7.3 (pages 156-160). In particular, note that the definite integral enjoys the same linearity properties that the derivative does, in addition to some others. In the applications with velocity functions, pay particular attention to the distinction between distance traveled and net distance traveled.

Watch this video until 30:00.

Work through problems 1-6 for Exercises 7.3. When you are done, check your answers against Appendix A.

- 7.4: Integration by Substitution
Read Section 8.1 (pages 161-166). This section explains the process of taking the integral of slightly more complicated functions. We do so by implementing a "change of variables," or rewriting a complicated integral in terms of elementary functions that we already know how to integrate. Simply speaking, integration by substitution is merely the act of taking the chain rule in reverse.

Watch this video from 30:00 to the end.

Work through problems 5-19 for Exercise 8.1. When you are done, check your answers against Appendix A.

- 7.5: Integration of Transcendental Functions
A transcendental number is a number that is not the root of any integer polynomial. A transcendental function, similarly, is a function that cannot be written using roots and the arithmetic found in polynomials. We address exponential, logarithmic, and hyperbolic functions here, having covered the integration and differentiation of trigonometric functions previously.

- 7.5.1: Exponential Functions
Read Section 8.3 (pages 441 through 447). This chapter recaps the definition of the number e and the exponential function and its behavior under differentiation and integration.

Work through problems 1-12. When you are done, check your solutions against the answers provided.

- 7.5.2: Natural Logarithmic Functions
Read Section 8.5 (pages 454 through 459). This chapter reintroduces the natural logarithm (the logarithm with base e) and discusses its derivative and antiderivative. Recall that you can use these properties of the natural log to extrapolate the same properties for logarithms with arbitrary bases by using the change of base formula.

Watch "Video 31: The Natural Logarithmic Function" through the 5th slide (marked 5 of 8). Then watch "Video 32: The Exponential Function."

The first short video gives one definition of the natural logarithm and derives all the properties of the natural log from that definition. It shows examples of limits, curve sketching, differentiation, and integration using the natural log. We will return to this video later to watch the last three slides. The second video explains the number e, the exponential function and its derivative and antiderivative, curve sketching using the exponential function, and how to perform similar operations on power functions with other bases using the change of base formula.

Click "Index" button for Book II. Scroll down to "3. Transcendental Functions," and click button 137 (Logarithm, Definite Integrals). Do problems 1-10. If at any time a problem set seems too easy for you, feel free to move on.

- 7.5.3: Hyperbolic Functions
Watch these videos. The creator of the video pronounces "sinh" as "chingk." The more usual pronunciation is "sinch."

Read Section 8.4 (pages 449 through 453). In this chapter, you will learn the definitions of the hyperbolic trig functions and how to differentiate and integrate them. The chapter also introduces the concept of capital accumulation.

Work through each of the sixteen examples on the page. As in any assignment, solve the problem on your own first. Solutions are given beneath each example.

- Unit 8: Applications of Integration
In this unit, we will take a first look at how integration can and has been used to solve various types of problems. Now that you have conceptualized the relationship between integration and areas and distances, you are ready to take a closer look at various applications; these range from basic geometric identities to more advanced situations in physics and engineering.

**Completing this unit should take you approximately 19 hours.** - 8.1: The Area between Curves
Suppose you want to find the area between two concentric circles. How would you do this? Logic dictates that you subtract the area of the smaller circle from that of the larger circle. As this subunit will demonstrate, this method also works when you are trying to determine the area between curves.

Read Section 4.5 (pages 218 through 222).

Watch this lecture from 21:30 to the end. In this lecture, Dr. Jerison will explain how to calculate the area between two curves.

Click the "Index" button for Book II. Scroll down to "2. Applications of Integration," and click button 115 (Area between Curves I). Work on problems 6-13. Next, choose button 116 (Area between Curves II), and complete problems 4-10. If at any time a problem set seems too easy for you, feel free to move on.

Work on problems 1-3 and 6-9. When you have finished, scroll down to the bottom of the page; the answers to the problems are printed upside-down at the end of the exercises.

The point of this third assignment is for you to practice setting up and completing these problems without the graphical aids provided by the Temple University media; you will have to graph these curves for yourself in order to begin the problems.

- 8.2: Volumes of Solids
We often take basic geometric formulas for granted. (Have you ever asked yourself why the volume of a right cylinder is \( V=\pi rh^2 \)?) In this subunit, we will explore how some of these formulas were developed. The key lies in viewing solids as functions that revolve around certain lines. Consider, for example, a constant, horizontal line, and then imagine that line revolving around the x-axis (or any parallel line). The resulting shape is a right cylinder. We can find the volume of this figure by looking at infinitesimally thin slices and adding them all together. This concept enables us to calculate the volume of some extremely complex figures. In this subunit, we will learn how to do this in general; in the next, we will take a look at two conventional methods for doing so when the figure has rotational symmetry.

Watch "Video 27: Volumes I". This video explains how to use integral calculus to calculate the volumes of general solids.

Work through each of the three examples on the page. As in any assignment, solve the problem on your own first. Solutions are given beneath each example.

- 8.3: Volume of Solids of Revolution
When we are presented with a solid that was produced by rotating a curve around an axis, there are two sensible ways to take that solid apart: slice it thinly perpendicularly to the axis, into disks (or washers, if the solid had a hole in the middle), or peel layers from around the outside like the paper wrapper of a crayon. The latter method is known as the shell method and produces thin cylinders. In both cases, we find the area of the thin segments and add them up to find the volume; as usual, when we have infinitely many pieces, this addition is really integration.

- 8.3.1: Disks and Washers
Read Section 6.2 (pages 308 through 318).

Watch this video. Dr. Jerison elaborates on some tangential material for a few minutes in the middle, but returns to the essential material very quickly.

Click on the "Index" for Book II. Scroll down to "2. Applications of Integration," and click button 119 (Solid of Revolution - Washers). Work on problems 1-12. If at any time a problem set seems too easy for you, feel free to move on.

- 8.3.2: Cylindrical Shells
Click on the "Index" for Book II. Scroll down to "2. Applications of Integration," and click button 120 (Solid of Revolution - Shells). Work on problems 5-17. If at any time a problem set seems too easy for you, feel free to move on.

Work through the exercises using the method you feel is most appropriate.

- 8.4: Lengths of Curves
In this subunit, we will make use of another concept that you have known and understood for quite some time: the distance formula. If you want to estimate the length of a curve on a certain interval, you can simply calculate the distance between the initial point and terminal point using the traditional formula. If you want to increase the accuracy of this measurement, you can identify a third point in the middle and calculate the sum of the two resulting distances. As we add more points to the formula, our accuracy increases: the exact length of the curve will be the sum (i.e. the integral) of the infinitesimally small distances.

Read Section 6.3 (pages 319 through 325). This reading discusses how to calculate the length of a curve, also known as arc length. This includes calculating arc length for parametrically-defined curves.

Watch this lecture until 26:10. Lecture notes are available in PDF; the link is on the same page as the lecture.

Click on the "Index" for Book II. Scroll down to "2. Applications of Integration," and click button 125 (Arc Length). Work on all of the problems (1-9). If at any time a problem set seems too easy for you, feel free to move on.

- 8.5: Surface Areas of Solids
In this subunit, we will combine what we learned earlier in this unit. Though you might expect that calculating the surface area of a solid will be as easy as finding its volume, it actually requires a number of additional steps. You will need to find the curve-length for each of the slices we identified earlier and then add them together.

Read Section 6.4 (pages 327 through 335). In this beautiful presentation of areas of surfaces of revolution, the author again makes use of rigorously-defined infinitesimals, as opposed to limits. Recall that the approaches are equivalent; using an infinitesimal is the same as using a variable and then taking the limit as that variable tends to zero.

Watch this video from 26:10 to 40:35.

Work through each of the three examples on the page. As in any assignment, solve the problem on your own first. Solutions are given beneath each example.

- 8.6: Average Value of Functions
You probably learned about averages (or mean values) quite some time ago. When you have a finite number of numerical values, you add them together and divide by the number of values you have added. There is nothing preventing us from seeking the average of an infinite number of values (i.e. a function over a given interval). In fact, the formula is intuitive: we add the numbers using an integral and divide by the length of the interval.

Read Section 6.5 (pages 336-340).

Watch this video until 30:00. In this lecture, Jerison will explain how to calculate average values and weighted average values.

Click on the "Index" for Book II. Scroll down to "4. Assorted Application," and click button 124 (Average Value). Work on problems 3-11. If at any time a problem set seems too easy for you, feel free to move on.

- 8.7: Physical Applications
We will now apply what we have learned about integration to various aspects of science. You may know that in physics, we calculate work by multiplying the force of the work by the distance over which it is exerted. You may also know that density is related to mass and volume. But we now know that distance and volume are very much related to integration. In this subunit, we will explore these and other connections.

- 8.7.1: Distance
Read Section 9.2 (pages 192-194).

Work through each of the three examples on the page. As in any assignment, solve the problem on your own first. Solutions are given beneath each example.

- 8.7.2: Mass and Density
Read this section (pages 341-351).

- 8.7.3: Moments
Work through each of the four examples on the page. As in any assignment, solve the problem on your own first. Solutions are given beneath each example.

- 8.7.4: Work
Read Section 9.5 (pages 205 through 208). Work is a fundamental concept from physics roughly corresponding to the distance traveled by an object multiplied by the force required to move it that distance.

Work through each of the seven examples on the page. As in any assignment, solve the problem on your own first. Solutions are given beneath each example.