Saylor Academy

Unit 3 Activities

Time Advisory

This unit should take you approximately 65.25 hours to complete.

Subunit 3.1: 22.25 hours
Subunit 3.2: 4.75 hours
Subunit 3.3: 4.5 hours
Subunit 3.4: 3.25 hours
Subunit 3.5: 22.5 hours
Subunit 3.6: 8 hours

Learning Outcomes

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

Explain in various ways how the definite integral is equal to a limit of Riemann sums.
Explain the definite integral of a rate of change over an interval as the change of a quantity over the interval [integral from a to b of f'(x)dx = f(b) - f(a) ].
Use basic properties of definite integrals (such as additivity and linearity) to solve more complex integrals.
Find basic antiderivatives based on known derivatives of basic functions.
Find more complex antiderivatives by substitution of variables and changing of limits.
Use the FTC to evaluate definite integrals.
Use the FTC to represent a particular antiderivative both analytically and graphically.
Use appropriate integrals to model physical, biological, and economic scenarios.
Use appropriate definite integrals in other situations by setting up an approximating Riemann sum and taking its limit in: Finding the area of a region.
Finding the volume of a solid with known cross sections.
Finding the average value of a function.
Finding the distance travelled by a particle along a path.
Finding an accumulation function or value given a rate of change function.
Find specific antiderivatives given initial conditions (especially involving motion along a line).
Solve separable differential equations and use them in modeling (specifically studying y'= ky and exponential growth).
Use Riemann sums (left, right, and midpoint) to approximate definite integrals of functions represented algebraically, graphically, and by tables of values.
Use trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values.

3.1 Techniques of Antidifferentiation (Integration)

Antidifferentiation (integration) can model real-life applications to determine the relationship among functions or a group of functions. Integration rules can help you eventually solve problems in multiple dimensions. Antidifferentiation is the opposite operation of differentiation, and this allows you to find values, such as areas under a curve, that are challenging without these concepts.

Subunit 3.1 focuses on the various techniques of antidifferentiation (integration) used in calculus. The subunit starts with the basic rules of antidifferentiation and then moves on to specialized cases such as trigonometry, integration by parts, integration by substitution, and partial fractions. These methods are critical to understanding definite integrals and additional calculus-related applications.

3.1.1 Basic Antidifferentiation Rules

Explanation: Khan Academy's "Antiderivatives and Indefinite Integrals” a> (YouTube)

Instructions: Please click on the link above, and take notes as you watch the video. Listen to the presentation carefully until you are able to understand the basic rule of an antiderivative (indefinite integral).

Watching this lecture and taking notes should take approximately 15 minutes.

Explanation: Khan Academy's "Indefinite Integrals of X Raised to a Power” a> and "Antiderivative of Hairier Expression” a> (YouTube)

Instructions: Please click on the link above, and take notes as you watch the videos. Listen to the presentations carefully until you are able to understand the basic rule of an antiderivative (indefinite integral) raised to a power.

Watching these lectures and taking notes should take approximately 45 minutes.

Explanation: CK-12 Calculus: "Indefinite Integrals Calculus” a> (HTML)

Instructions: Please click on the link above, and read the material on indefinite integrals, stopping at "Review Questions.” This reading discusses eight basic indefinite integrals using the rules of derivatives to explain each step.

Reading and taking notes on this text should take approximately 45 minutes.

Explanation: Khan Academy's "Basic Trig and Exponential Antiderivatives” a> (YouTube)

Instructions: Please click on the link above, and take notes as you watch the video from the 1:52 mark to the end. Listen to the presentation carefully until you are able to understand the basic rule of an exponential antiderivative (indefinite integral).

Watching this lecture and taking notes should take approximately 15 minutes.

Explanation: Khan Academy's "Antiderivative of x^-1” a> (YouTube)

Watching this lecture and taking notes should take approximately 30 minutes.

Checkpoint: Wikibooks: "Calculus/Integration/Exercises” a> (YouTube)

Instructions: Please click on the link above, and complete exercises 1-5 and 10-12. These exercises will provide you with the opportunity to apply basic rules of integration. The solution for each problem can be found by clicking on the gray triangle beside each problem.

Completing this activity should take approximately 30 minutes.

Did I Get This? Activity: CK-12 Calculus: "Indefinite Integrals Calculus” a> (HTML)

Instructions: Please click on the link above, and complete review questions 1-10 toward the bottom of the page. These exercises will provide you with the opportunity to find antiderivatives as well as indefinite integrals of various functions. The solutions to these problems are located here.

Completing this activity should take approximately 45 minutes.

Reading: Math Insight: "Developing Intuition about the Indefinite Integral” a> (HTML and You Tube)

Instructions: Please click on the link above, and read the material on describing the indefinite integral. Please also run each of the applets to see examples. This literacy component will allow you to explore a further understanding of the indefinite integral.

Reading and taking notes on this text should take approximately 15 minutes.

Standards Addressed (Common Core):

CCSS.ELA.Literacy.RST.11-12.3
CCSS.ELA.Literacy.RST.11-12.7

3.1.2 Integration by Substitution

Explanation: Khan Academy's "U-Substitution,” a> "U-Substitution Example 2,” a> and "U-Substitution Example 3” a> (YouTube)

Instructions: Please click on the links above, and take notes as you watch the videos. Listen to the presentations carefully until you are able to understand the basic rules of integration by substitution.

Watching these lectures and taking notes should take approximately 45 minutes.

Explanation: CK-12 Calculus: "Integration by Substitution” (HTML and YouTube)

Instructions: Please click on the link above, and read the material on trigonometric integration, stopping at "Trigonometric Integrands.” Then, continue on to watch the video entitled "Math Video Tutorials by James Sousa, Integration by Substitution, Part 1 of 2.” This video discusses the process of integration by substitution (or the u-substitution method).

Reading, taking notes, and viewing the video should take approximately 45 minutes.

Checkpoint: CK-12 Calculus: "Integration by Substitution” (HTML)

Instructions: Please click on the link above, and complete review questions 1-7 toward the bottom of the page. These exercises will provide you with the opportunity to find antiderivatives as well as indefinite integrals of various functions using the substitution method. The solutions to these problems are located here.

Completing this activity should take approximately 45 minutes.

Did I Get This? Activity: CK-12 Calculus: "Integration by Substitution” (HTML)

Instructions: Please click on the link above, and complete review questions 1, 3, and 7-9 toward the bottom of the page. These exercises will provide you with the opportunity to find indefinite integrals of various functions using the substitution method. The solutions to these problems are located here.

Completing this activity should take approximately 30 minutes.

Reading: The City of University of New York's Mathematics Blog: "My Monumental Ignorance: Proofs I Wish I Knew, and the Challenge of Negativity” a> (HTML)

Instructions: Please click on the link above, and read the material on various functions and integration. This literacy component will allow you to explore a further understanding of function properties and integration. The blog also contains information on how a specific function cannot be integrated. Write a one-page summary that discusses the key components of the article.

Reading and taking notes on this text should take approximately 30 minutes.

Standards Addressed (Common Core):

3.1.3 Trigonometry Integration

3.1.3.1 Trigonometry Integrands

Explanation: CK-12 Calculus: "Integration by Substitution” (HTML and YouTube)

Instructions: Please click on the link above, and read the material on trigonometric integration, starting at "Trigonometric Integrands” and stopping at "Using Substitution on Definite Integrals.” Then, continue on to watch the video entitled "Math Video Tutorials by James Sousa, Integration by Substitution, Part 2 of 2.” This video discusses the process of integration by substitution (or the u-substitution method) and the basic trigonometric integrals.

Reading, taking notes, and viewing this video should take approximately 45 minutes.

Checkpoint: Wikibooks: "Calculus/Integration/Exercises” a> (HTML)

Instructions: Please click on the link above, and complete exercises 6-9. These exercises will provide you with the opportunity to apply basic rules of integration. The solution for each problem can be found by clicking on the gray triangle beside each problem.

Completing this activity should take approximately 30 minutes.

Did I Get This? Activity: CK-12 Calculus: "Integration by Substitution” (HTML)

Instructions: Please click on the link above, and complete review questions 8-12 toward the bottom of the page. These exercises will provide you with the opportunity to find indefinite integrals of trigonometric functions using the substitution method. The solutions to these problems are located here.

Completing this activity should take approximately 45 minutes.

3.1.3.2 Trigonometry Integrals

Explanation: CK-12 Calculus: "Trigonometric Integrals” (HTML and YouTube)

Instructions: Please click on the link above, and read the material on trigonometric integration, stopping at "Multimedia Links.” Then, continue on to watch all videos within the section. This material discusses the process of integration of powers of sines and cosines as well as secants and tangents.

Reading, taking notes, and viewing the videos should take approximately 2 hours.

Checkpoint: CK-12 Calculus: "Trigonometric Integrals” (HTML and YouTube)

Instructions: Please click on the link above, and complete review questions 1-7 toward the bottom of the page. These exercises will provide you with the opportunity to find indefinite integrals of trigonometric functions using the substitution method and trigonometric integrals. The solutions to these problems are located here.

Completing this activity should take approximately 1 hour.

3.1.3.3 Trigonometry Substitutions

Explanation: YouTube: North Carolina School of Science and Mathematics: "Trigonometric Substitutions” a> (YouTube)

Instructions: Please click on the link above, and take notes as you watch the video. Listen to the presentation carefully until you are able to understand how to apply rules of trigonometric substitutions to find specific indefinite integrals.

Watching this lecture and taking notes should take approximately 45 minutes.

Explanation: CK-12 Calculus: "Trigonometric Substitutions” (HTML and YouTube)

Instructions: Please click on the link above, and read the material on trigonometric substitutions, stopping at "Multimedia Links.” Then, continue on to watch both videos within the section. This material discusses the process of integration using trigonometric integrals, substitution method for integrals, and other techniques.

Reading, taking notes, and viewing the videos should take approximately 2 hours.

Checkpoint: CK-12 Calculus: "Trigonometric Substitutions” (HTML)

Instructions: Please click on the link above, and complete review questions 1-7 toward the bottom of the page. These exercises will provide you with the opportunity to find indefinite integrals using trigonometric integrals, substitution method for integrals, and other techniques. The solutions to these problems are located here.

Completing this activity should take approximately 1 hour.

3.1.4 Integration by Parts

Explanation: YouTube: North Carolina School of Science and Mathematics: "Integration by Parts” (YouTube)

Instructions: Please click on the link above, and take notes as you watch the video. Listen to the presentation carefully until you are able to understand how to apply the formula for integration by parts to various functions.

Watching this lecture and taking notes should take approximately 1 hour.

Explanation: CK-12 Calculus: "Integration by Parts” (HTML and YouTube)

Instructions: Please click on the link above, and read the material on integration by parts, stopping at "Multimedia Links.” Then, continue on to watch all videos within the section. This material discusses the process of integration by parts for various functions.

Reading, taking notes, and viewing the videos should take approximately 2 hours.

Checkpoint: CK-12 Calculus: "Integration by Parts” (HTML)

Instructions: Please click on the link above, and complete review questions 1-9 toward the bottom of the page. These exercises will provide you with the opportunity to find indefinite integrals using integration by parts. The solutions to these problems are located here.

Completing this activity should take approximately 1 hour.

3.1.5 Integration by Partial Fractions

Explanation: YouTube: North Carolina School of Science and Mathematics: "Method of Partial Fractions” (YouTube)

Watching this lecture and taking notes should take approximately 45 minutes.

Explanation: CK-12 Calculus: "Integration by Partial Fractions” (HTML and YouTube)

Instructions: Please click on the link above, and read the material on integration by partial fractions, stopping at "Multimedia Links.” Then, continue on to watch the first two videos within the section, omitting the "MIT Courseware” video. This material discusses the process of integration by using partial fractions.

Reading, taking notes, and viewing the videos should take approximately 1.25 hours.

Checkpoint: CK-12 Calculus: "Integration by Partial Fractions” (HTML)

Instructions: Please click on the link above, and complete review questions 1-10 toward the bottom of the page. These exercises will provide you with the opportunity to find indefinite integrals using integration by using partial fractions. The solutions to these problems are located here.

Completing this activity should take approximately 1.5 hours.

3.2 Interpretations and Properties of Definite Integrals

Definite integrals allow us to find the exact area underneath a curve. Without these integrals, it would be very difficult to even estimate the area under a curve that is not a linear function. The rules of integration allow us to find the area underneath any function.

Subunit 3.2 focuses on finding definite integrals for various functions. The rules of integration learned in subunit 3.1 will be explored further as we work to find the exact values of integrals over a given region.

Explanation: YouTube: North Carolina School of Science and Mathematics: "Definite Integral Interpretation” (YouTube)

Instructions: Please click on the link above, and take notes as you watch the video. Listen to the presentation carefully until you are able to understand how to apply the rules of integration and then find the area of a specific region.

Watching this lecture and taking notes should take approximately 30 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

Explanation: CK-12 Calculus: "Evaluating Definite Integrals” (HTML)

Instructions: Please click on the link above, and read the material on evaluating definite integrals, stopping at "Review Questions.” This reading discusses the process of finding definite integrals and incorporates the rules of integrals to solve problems.

Reading and taking notes on this text should take approximately 30 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

Explanation: Khan Academy's "Evaluating Simple Definite Integral” and "Definite Integrals and Negative Area” (YouTube)

Instructions: Please click on the links above, and take notes as you watch the videos. Listen to the presentations carefully until you are able to apply the rule of integrations to find a definite integral.

Watching these lectures and taking notes should take approximately 45 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

Checkpoint: CK-12 Calculus: "Evaluating Definite Integrals” (HTML)

Instructions: Please click on the link above, and complete review questions 1-8 and 10 toward the bottom of the page. These exercises will provide you with the opportunity to find definite integrals using integration rules. The solutions to these problems are located here.

Completing this activity should take approximately 1 hour.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

Did I Get This? Activity: CK-12 Calculus: "Integration by Substitution” (HTML)

Instructions: Please click on the link above, and complete review questions 2, 4, 6, and 10 toward the bottom of the page. These exercises will provide you with the opportunity to find definite integrals of various functions using the various integral rules. The solutions to these problems are located here.

Completing this activity should take approximately 30 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

Did I Get This? Activity: CK-12 Calculus: "Integration by Substitution” (HTML)

Instructions: Please click on the link above, and complete review questions 13-15 toward the bottom of the page. These exercises will provide you with the opportunity to find definite integrals of various functions using the various integral rules. The solutions to these problems are located here.

Completing this activity should take approximately 30 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

Did I Get This? Activity: CK-12 Calculus: "Integration by Parts” (HTML)

Instructions: Please click on the link above, and complete review questions 10-11 toward the bottom of the page. These exercises will provide you with the opportunity to find definite integrals of various functions using the various integral rules. The solutions to these problems are located here.

Completing this activity should take approximately 15 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

Did I Get This? Activity: CK-12 Calculus: "Integration by Partial Fractions” (HTML)

Instructions: Please click on the link above, and complete review questions 5-6 toward the bottom of the page. These exercises will provide you with the opportunity to find definite integrals of various functions using the various integral rules. The solutions to these problems are located here.

Completing this activity should take approximately 15 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

3.3 Fundamental Theorem of Calculus

How are a derivative and an integral related? The fundamental theorem of calculus ties these concepts together and helps us to quickly find the area underneath the curve of a function. This theorem allows us to perform complex calculations at a faster rate using a very simple rule.

The fundamental theorem of calculus provides us with another approach to finding definite integrals and is used when multiple functions are involved in finding a specific area.

Explanation: Khan Academy's "Fundamental Theorem of Calculus” (YouTube)

Instructions: Please click on the link above, and take notes as you watch the video. Listen to the presentation carefully until you are able to understand how integrals and derivatives are use to prove the fundamental theorem of calculus.

Watching this lecture and taking notes should take approximately 30 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

Explanation: CK-12 Calculus: "The Fundamental Theorem of Calculus” (HTML and YouTube)

Instructions: Please click on the link above, and read the material on the fundamental theorem of calculus, stopping at "Multimedia Link.” Please continue on to watch the video in this section. This material discusses the process of using the fundamental theorem of calculus to evaluate definite integrals.

Reading, taking notes, and viewing the video should take approximately 45 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

Explanation: Khan Academy's "Applying the Fundamental Theorem of Calculus” and "Swapping the Bounds for Definite Integral” (YouTube)

Instructions: Please click on the links above, and take notes as you watch the videos. Listen to the presentations carefully until you are able to apply the rule of integrations to find a definite integral.

Watching these lectures and taking notes should take approximately 30 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

Checkpoint: CK-12 Calculus: "The Fundamental Theorem of Calculus” (HTML)

Instructions: Please click on the link above, and complete review questions 1-10 toward the bottom of the page. These exercises will provide you with the opportunity to apply the fundamental theorem of calculus to evaluate definite integrals. The solutions to these problems are located here.

Completing this activity should take approximately 1.25 hours.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

Explanation: Khan Academy's "2011 Calculus AB Free Response #4a,” "2011 Calculus AB Free Response #4b,” "2011 Calculus AB Free Response #4c,” and "2011 Calculus AB Free Response #4d” (YouTube)

Instructions: Please click on the links above, and take notes as you watch the videos. Listen to the presentations carefully until you are able to understand all of the parts of the problem from the Calculus AB Exam.

Watching these lectures and taking notes should take approximately 30 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

3.4 Improper Integrals

What happens when an integral is undefined? What can we use when one of the values on the interval is infinite? Improper integrals help us define integrals when the fundamental theorem of calculus cannot be used because the interval is not continuous. Since not all functions are continuous within each interval, other approaches need to be used to find the value of the definite integral.

This subunit defines how to find the values of improper integrals using a variety of techniques. One approach is used when infinity is contained in the interval and limits are used. Another approach is used when there is discontinuity within the integral.

Explanation: YouTube: North Carolina School of Science and Mathematics: "Improper Integrals” (YouTube)

Instructions: Please click on the link above, and take notes as you watch the video. Listen to the presentation carefully until you are able to understand the concept of integrals involving infinity.

Watching this lecture and taking notes should take approximately 30 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

Explanation: CK-12 Calculus: "Improper Integrals” (HTML and YouTube)

Instructions: Please click on the link above, and read the material on improper integrals, stopping at "Multimedia Links.” Then continue on to watch both videos in this section. This material discusses the process of using integration techniques to solve improper integrals.

Reading, taking notes, and viewing the videos should take approximately 1 hour.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

Explanation: Khan Academy's "Introduction to Improper Integrals” and "Improper Integral with Two Infinite Bounds” (YouTube)

Instructions: Please click on the links above, and take notes as you watch the videos. Listen to the presentations carefully until you are able to apply the rules of integrals to solve a problem with an improper integral.

Watching these lectures and taking notes should take approximately 1 hour.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

Checkpoint: CK-12 Calculus: "Improper Integrals” (HTML)

Instructions: Please click on the link above, and complete review questions 1-6 toward the bottom of the page. These exercises will provide you with the opportunity to solve problems that contain improper integrals. The solutions to these problems are located here.

Completing this activity should take approximately 45 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

3.5 Applications of Integrals

Applications of integrals are plentiful and allow one to find calculations to complex functions. With rules of integrals, we can find the area under a curve, and by expanding this rule, we can find the volume between curves. One example is determining the amount of work required to lift a bucket attached to a cable. Finding distance, velocity, and acceleration of an object being shot in the area is another application of integrals.

Subunit 3.5 explores the many applications of integrals. These include the area between curves, the volumes of various figures, the length of plane curves, and the area of various surfaces. There are also many applications within other disciplines that will be explored in this subunit.

3.5.1 Area between Two Curves

Explanation: YouTube: North Carolina School of Science and Mathematics: "Area between Two Curves” (YouTube)

Instructions: Please click on the link above, and take notes as you watch the video. Listen to the presentation carefully until you are able to understand how to find the area between two curves.

Watching this lecture and taking notes should take approximately 45 minutes.

Explanation: CK-12 Calculus: "Area between Two Curves” (HTML and YouTube)

Instructions: Please click on the link above, and read the material on the area between two curves, stopping at "Multimedia Links.” Then continue on to watch the two videos in this section. This material discusses the process of finding the area between two curves with respect to the x-axis and y-axis.

Reading and taking notes on this text should take approximately 1.25 hours.

Checkpoint: CK-12 Calculus: "Area between Two Curves” (HTML)

Instructions: Please click on the link above, and complete review questions 1-10 toward the bottom of the page. These exercises will provide you with the opportunity to solve problems using integrals to find the area between two curves. The solutions to these problems are located here.

Completing this activity should take approximately 1.5 hours.

Explanation: Khan Academy's "2011 Calculus AB Free Response #3 (a & b)” and "2011 Calculus AB Free Response #3 (c)” (YouTube)

Instructions: Please click on the links above, and take notes as you watch the videos. Listen to the presentations carefully until you are able to understand all of the parts of the problem from the Calculus AB Exam.

Watching these lectures and taking notes should take approximately 30 minutes.

3.5.2 Volumes

3.5.2.1 Disc, Washer, and Shell Methods Around an Axis

Explanation: YouTube: North Carolina School of Science and Mathematics: "Volumes Using Discs and Washers” (YouTube)

Instructions: Please click on the link above, and take notes as you watch the video, stopping at the 15:33 mark. Listen to the presentation carefully until you are able to find the volume of figures using disks and washers.

Watching this lecture and taking notes should take approximately 45 minutes.

Explanation: CK-12 Calculus: "Volumes” (HTML)

Instructions: Please click on the link above, and read the material on volumes, stopping at "Volume by Cylindrical Shells.” This reading discusses the process of finding the volume of objects using the disk and washer methods.

Reading and taking notes on this text should take approximately 30 minutes.

Explanation: Khan Academy's "Disk Method Around X-Axis” and "Disk Method Around Y-Axis” (YouTube)

Instructions: Please click on the links above, and take notes as you watch the videos. Listen to the presentations carefully until you are able to find the volume of a figure rotated around an axis using the disk method.

Watching these lectures and taking notes should take approximately 1 hour.

Explanation: Khan Academy's "Disk Method (Washer Method) for Rotation Around X-Axis” (YouTube)

Instructions: Please click on the link above, and take notes as you watch the video. Listen to the presentation carefully until you are able to find the volume of a figure rotated around an axis using the washer method.

Watching this lecture and taking notes should take approximately 30 minutes.

Explanation: CK-12 Calculus: "Volumes” (HTML)

Instructions: Please click on the link above, and read the material on trigonometric integration, starting at "Volumes by Cylindrical Shells” and stopping at "Multimedia Links.” This material discusses the process of finding the volume of objects using the cylindrical shell method.

Reading and taking notes on this text should take approximately 30 minutes.

Explanation: Khan Academy's "Shell Method for Rotating Around Vertical Line” and "Shell Method for Rotating Around Horizontal Line” (YouTube)

Watching these lectures and taking notes should take approximately 1 hour.

Explanation: CK-12 Calculus: "Volumes” (HTML and YouTube)

Instructions: Please click on the link above, and read the material on trigonometric integration, stopping at "Multimedia Links.” Then go on to watch all three videos in this section. This material discusses the process of finding the volume of objects using the disk and shell method to obtain the same results

Reading, taking notes, and watching the videos should take approximately 1 hour.

Checkpoint: CK-12 Calculus: "Volumes” (HTML and YouTube)

Instructions: Please click on the link above, and complete review questions 1-12 toward the bottom of the page. These exercises will provide you with the opportunity to find the object of volumes using the disk, washer, and shell methods. The solutions to these problems are located here.

Completing this activity should take approximately 2.25 hours.

3.5.2.2 Disc, Washer, and Shell Methods Around a Nonaxis Line

Explanation: YouTube: North Carolina School of Science and Mathematics: "Volumes Using Discs and Washers” (YouTube)

Instructions: Please click on the link above, and take notes as you watch the video, starting at the 15:33 mark and watching until the end. Listen to the presentation carefully until you are able to find the volume of figures using discs and washers rotating around a nonaxis line.

Watching this lecture and taking notes should take approximately 15 minutes.

Explanation: Khan Academy's "Disk Method Rotation Around Horizontal Line” and "Disk Method Rotation Around Vertical Line” (YouTube)

Watching these lectures and taking notes should take approximately 1 hour.

Explanation: Khan Academy's "Washer Method Rotating Around Non-Axis,” "Part 2 of Washer for Non Axis Rotation,” and "Washer or Ring Method for Vertical Line Rotation” (YouTube)

Watching these lectures and taking notes should take approximately 1.5 hours.

Explanation: Khan Academy's "Shell Method with Two Functions of X” and "Shell Method with Two Functions of Y” (YouTube)

Watching these lectures and taking notes should take approximately 45 minutes.

3.5.3 The Length of a Plane Curve

Explanation: CK-12 Calculus: "The Length of a Plane Curve” (HTML and YouTube)

Instructions: Please click on the link above, and read the material on the area between two curves, stopping at "Multimedia Links.” Then move on to watch the two videos in this section. This material discusses the process of finding the length of a plane curve for a given function.

Reading, taking notes, and viewing the videos should take approximately 45 minutes.

Checkpoint: CK-12 Calculus: "The Length of a Plane Curve” (HTML)

Instructions: Please click on the link above, and complete review questions 1-4 toward the bottom of the page. These exercises will provide you with the opportunity to solve problem using integrals to find the length of a plane curve. The solutions to these problems are located here.

Completing this activity should take approximately 30 minutes.

3.5.4 Area of a Surface of Revolution

Explanation: CK-12 Calculus: "Area of a Surface of Revolution” (HTML and YouTube)

Reading, taking notes, and viewing the videos should take approximately 1 hour.

Checkpoint: CK-12 Calculus: "Area of a Surface of Revolution” (HTML)

Instructions: Please click on the link above, and complete review questions 1-6 toward the bottom of the page. These exercises will provide you with the opportunity to solve problems using integrals to find the area of a surface of revolution. The solutions to these problems are located here.

Completing this activity should take approximately 30 minutes.

3.5.5 Other Applications

Explanation: CK-12 Calculus: "Applications from Physics, Engineering, and Statistics” (HTML and YouTube)

Instructions: Please click on the link above, and read the material on the area between two curves, stopping at "Multimedia Links.” Then move on to watch all videos in this section, except the video entitled "Khan Academy, Centripetal Acceleration.” This material discusses the uses of integral applications from various disciplines.

Reading, taking notes, and viewing the videos should take approximately 2.5 hours.

Checkpoint: CK-12 Calculus: "Applications from Physics, Engineering, and Statistics” (HTML and YouTube)

Instructions: Please click on the link above, and complete review questions 1-10 toward the bottom of the page. These exercises will provide you with the opportunity use integrals to solve problems from other disciplines. The solutions to these problems are located here.

Completing this activity should take approximately 2 hours.

3.6 Numerical Approximations to Definite Integrals

There are times when the antiderivative is very challenging to find or may not even exist, which means the definite integral cannot be evaluated. What can one do in this case when the integral cannot be found? Three approaches can be taken to find the approximate value of the region under a curve.

Subunit 3.6 looks at various ways to use numerical approximations to find definite integrals. The Riemann sum, trapezoidal rule, and Simpson's rule can all be used to find an accurate approximation of an area under a curve.

3.6.1 Riemann Sums

Explanation: Khan Academy's "Simple Riemann Approximation Using Rectangles” and "Generalizing a Left Riemann Sum with Equally Spaced Rectangles” (YouTube)

Instructions: Please click on the links above, and take notes as you watch the videos. Listen to the presentations carefully until you are able to use Riemann approximation to estimate areas under curves.

Watching these lectures and taking notes should take approximately 1 hour.

Explanation: CK-12 Calculus: "Definite Integrals” (HTML and YouTube)

Instructions: Please click on the link above, and read the material on the area between two curves, stopping at "Multimedia Link.” Then move on to watch the two videos in this section. This material discusses how to use Riemann sums to approximate the area under a curve.

Reading, taking notes, and viewing the videos should take approximately 1.25 hours.

Checkpoint: CK-12 Calculus: "Definite Integrals” (HTML)

Instructions: Please click on the link above, and complete review questions 1-10 toward the bottom of the page. These exercises will provide you with the opportunity to use Riemann sums to solve various integral problems. The solutions to these problems are located here.

Completing this activity should take approximately 1.25 hours.

Reading: Math Insight: "Calculating the Area Under a Curve Using Riemann Sums” (HTML)

Instructions: Please click on the link above, and read the material on Riemann sums, stopping at "Summary of Questions.” In this reading, also run each of the applets to provide examples of calculating the area. This literacy component will allow you to explore an application of Riemann sums. Information in this reading pertains to finding the area under the curve using Riemann sums. Write a one-page summary that discusses the key components of the article.

Reading and taking notes on this text should take approximately 45 minutes.

Standards Addressed (Common Core):

3.6.2 Trapezoidal Sums

Explanation: Khan Academy's "Trapezoidal Approximation of Area Under Curve” (YouTube)

Instructions: Please click on the link above, and take notes as you watch the video. Listen to the presentations carefully until you are able to use trapezoidal approximation to estimate areas under curves.

Watching this lecture and taking notes should take approximately 30 minutes.

Explanation: CK-12 Calculus: "Numerical Integration” (HTML and YouTube)

Instructions: Please click on the link above, and read the material on the area between two curves, stopping at "Multimedia Links.” Then move on to watch the two videos in this section. This material discusses how to use the trapezoidal and Simpson rules to approximate the area under a curve.

Reading, taking notes, and viewing the videos should take approximately 1.25 hours.

Checkpoint: CK-12 Calculus: "Numerical Integration” (HTML)

Instructions: Please click on the link above, and complete review questions 1-10 toward the bottom of the page. These exercises will provide you with the opportunity to use the trapezoidal and Simpson rules to solve various integral problems. The solutions to these problems are located here.

Completing this activity should take approximately 1 hour.

Explanation: Khan Academy's "2011 Calculus AB Free Response #2 (a & b)” and "2011 Calculus AB Free Response #2 (c & d)” (YouTube)

Watching these lectures and taking notes should take approximately 1 hour.

Last modified: Tuesday, June 16, 2015, 1:28 PM

K12MATH013: Calculus AB

Unit 3 Activities

Time Advisory

Learning Outcomes