Integrals Teacher Resources

Find Integrals lesson plans and worksheets
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Introduction to Calculus

For Teachers 11th - Higher Ed Standards
This heady calculus text covers the subjects of differential and integral calculus with rigorous detail, culminating in a chapter of physics and engineering applications. A particular emphasis on classic proof meshes with modern graphs,...
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Historical Reflections on the Fundamental Theorem Of (Integral) Calculus

For Teachers 11th - 12th
Highlight the process of mathematical discovery for your classes. This lesson examines the development of the Fundamental Theorem of Calculus over time. Understanding the history of the theorem helps create a better understanding of the...
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The Fundamental Theorems of Calculus

For Teachers 11th - Higher Ed
Are your calculus pupils aware that they are standing on the shoulders of giants? This instructional activity provides a big picture view of the connection between differential and integral calculus and throws in a bit of history, as...
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Math Handbook: Calculus

For Students 10th - 12th
If only there was a reference that listed all the formulas covered in calculus. Luckily, there is. This comprehensive 197-page handbook provides formulas and explanations for all topics in AP Calculus or in a standard college calculus...
7:54
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Properties of the Definite Integral

For Students 11th - Higher Ed
What are the properties that can be associated with the definite integral? The AP Calculus video lists nine properties of the definite integral and provides explanations of why the properties hold for integrals in language that is simple...
7:42
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Calculus, What Is It Good For?

For Students 11th - Higher Ed
Explore the meaning of differentiation and integration. A video lesson describes in detail what differentiating and integrating an equation means using the momentum and kinetic energy formulas. Progressing from a linear to a quadratic...
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The AP Calculus Problem Book

For Students 11th - Higher Ed Standards
Are you always looking for extra problem resources to incorporate in your classes?  A set of problems for a full AP Calculus course may be all you need! Each unit has a full set of questions and a sample test.
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Indefinite Integrals

For Teachers 11th - Higher Ed Standards
The inverse of matter is anti-matter, so the inverse of a derivative is an antiderivative. The lesson introduces antiderivatives and indefinite integrals to the class along with the notation for integrals. Groups work together to find...
1 In 1 Collection
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Derivative and Integral Rules

For Teachers 10th - Higher Ed
Your calculus learners will enjoy this handy reference sheet containing the most used derivative and integral formulas.
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Properties of Definite Integrals

For Students 11th - Higher Ed
Close your eyes and visualize a definite integral. An interactive provides a visualization of a function and definite integral on a closed interval. Pupils move one point of the interval in either the positive or negative direction....
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Study Guide for the Advanced Placement Calculus AB Examination

For Teachers 11th - 12th Standards
Is this going to be on the test? A calculus study guide provides an organized list of important topics and a few examples with answers. The topics include elementary functions, limits, differential calculus, and integral calculus,...
20:45
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Integration and the Fundamental Theorem of Calculus | Essence of Calculus, Chapter 8

For Students 11th - Higher Ed
Integrate a great resource into your lesson plans. The eighth video in an 11-part series explains integration and the relationship between derivatives and integrals. The relationship between a car's velocity and its travel distance helps...
2:54
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Solving Integrals Using Substitution

For Students 11th - Higher Ed
The concept of solving integrals by substitution with trigonometric functions is modeled by a video designed for AP Calculus classes. The narrator works an example problem using substitution and shows that careful substitution makes...
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The Fundamental Theorem of Calculus

For Teachers 10th - 12th Standards
Practice in the mechanics of using substitution in integration is often hard to come by in enough quantity for learners to really wrap their heads around the nuances of the technique. This worksheet set with solution key and worked...
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Integral Calculus

For Teachers 10th - Higher Ed
From derivatives to antiderivatives and back again. Building on the second instructional activity of the three-part series covering functions, learners explore the concept of an antiderivative. They connect the concept to the graph of...
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Integration by Substitution

For Teachers 11th - Higher Ed
Looking for help in introducing integration by substitution to your calculus students?  Here is a lesson that walks a learner through a review of the change rule, and then by starting with more recognizable derivatives, it helps them...
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Calculus

For Students 11th - Higher Ed Standards
Everything you wanted to know about calculus—and more! The resource is a complete Calculus textbook with explanations, examples, and practice problems. 
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Definite Integrals and Area Under a Curve

For Teachers 11th - Higher Ed
Students explore the concept of definite integrals.  For this definite integrals lesson, students find the area between two curves.  Students use the Ti-Nspire to find the definite integrals of curves such as sine and cosine.
9:00
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Natural Log Function-Integration

For Teachers 11th - Higher Ed Standards
One of the strongest tools in integration is u-substitution, particularly when working with natural logs. In this step-by-step video presentation, a pleasant-voiced instructor walks through a number of integration problems. She begins...
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A.P. Calculus Integration Test

For Students 12th - Higher Ed
In this A.P. Calculus activity, students complete a sixteen question test covering trigonometric integration, area under a curve, differential equations, and slope fields.  Some of the problems are multiple choice, while others are...
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Calculus Assignment #1: Integrals

For Students 10th - 12th
In this calculus learning exercise, students evaluate given integrals. Problems contain exponents, square roots, and complex fractions. This two-page learning exercise contains 16 multi-step problems. Answers are provided on page two.
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Definite Integral

For Teachers 9th - 12th Standards
Learners investigate the intervals represented by a function in this calculus lesson. They decide what interval of the function will be positive, negative or zero. They are then given graphs of functions and asked to analyze it.
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Integrals

For Students 12th - Higher Ed
For this integrals worksheet, students solve and complete 22 various types of problems. First, they evaluate each of the listed integrals by using substitution. Then, students solve each of the given differential equations when given the...
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Integrals

For Students 12th - Higher Ed
In this integrals worksheet, students solve and complete 22 various types of problems. First, they evaluate each of the listed integrals on the sheet. Then, students solve each of the given differential equations by substituting the...
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Integrals

For Students 12th - Higher Ed
In this integrals activity, learners solve and complete 22 different types of problems. First, they evaluate each of the listed integrals provided on the sheet. Then, students solve each of the differential equations. They also graph a...
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Calculus Grapher

For Teachers 8th - 12th Standards
Learners draw a graph of any function to see its derivative and integral in this simple, yet informative, simulation. Other options include manipulating the scale, the shape of the graph, and the angle. 
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Calculus - Early Transcendentals

For Teachers 9th - Higher Ed Standards
This textbook takes the learner from the basic definition of slope through derivatives, integrals, and vector multivariable calculus. Each section is composed primarily of examples, with theoretical introductions and explanations in...
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InteGreat

For Teachers 9th - Higher Ed Standards
Hands-on investigation of Riemann sums becomes possible without intensive arithmetic gymnastics with this interactive lesson plan. Learners manipulate online graphing tools to develop and test theories about right, left, and midpoint...
10:08
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Integrals

For Students 9th - 12th Standards
How can you best teach your class to gauge distance when several other key factors are already in place? Pupils learn to use integrals to determine distance from a problem that includes multiple other key points of information....
1:46
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The Second Fundamental Theorem of Calculus

For Students 11th - Higher Ed
How many fundamental theorems are there? The Second Fundamental Theorem of Calculus is also known as the second part of the Fundamental Theorem of Calculus. Here's a video that introduces the theorem as the method to evaluate a definite...
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Integration

For Teachers 12th
Twelfth graders investigate integration.  In this Calculus activity, 12th graders explore indefinite and definite integrals.  Students use technology (TI-89) to verify the solutions to problems requiring integration by parts.
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Integration

For Teachers 11th - 12th
Students integrate integrals.  In this integrating integrals lesson, students integrate definite and indefinite integrals.  Students use their Ti-89 to find the integrals of cubic, inverse, and trigonometric function graphs on a given...
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Calculus Assignment #5: Integrals

For Students 11th - 12th
In this calculus assignment learning exercise, students evaluate given integrals. Problems contain square roots, complex fractions, and exponents. This two-page learning exercise contains 14 problems. Answers are provided on page two...
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Proof That One Equals Zero (Using Calculus)

For Students 11th - 12th
For this proofs worksheet, students evaluate the integral of 1 function. Students use uv-substitution to prove 0=1.
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Calculus Volume 1

For Students 11th - Higher Ed
Differentiate between calculus concepts. The beginning of a helpful calculus resource begins with a short review of functions and moves right into limits to define derivatives. The six-chapter resource continues with applications of...
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Exploration 3: Introduction to Definite Integrals

For Students 11th - 12th
In this integral worksheet, students estimate the definite integral and find the derivative of functions. They graph functions and determine the range of the velocity. This three-page worksheet contains approximately 40 problems.
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2017 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
What is the answer to this question? Pupils use six questions to practice for the AP Calculus AB test. The review questions are divided in two sections, calculator and non-calculator. Additional resources allow teachers to determine how...
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Lesson 33- Integration by Substitution continued

For Students 11th - 12th
Humongous Calculus problem of the day - "Two hikers begin ..." Use the properties of indefinite integers and indefinte integrals to solve this problem.  Six practice problems are included with space to do the work, as well as a...
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Calculus Practice Test: Velocity, Functions

For Students 10th - 11th
In this calculus worksheet, students observe graphs and identify the limits of the functions listed in the graph. They determine the definite integrals and derivatives. Students use the trapezoid rule to estimate distance. This...
5:59
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Volumes by Cylindrical Shells

For Students 11th - Higher Ed
Shells sometimes work better than washers. The AP Calculus video presents an example of finding the volume of a rotational solid by using the shell method. The method uses the integral of the lateral surface area of a cylinder as opposed...
3:23
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Volumes: Solids That are Not Solid of Revolution

For Students 11th - Higher Ed
Viewers learn how to calculate the volume when the cross section is not a circle by using the same principle as if it was a circle. The video goes through an example of finding the volume of a figure where the cross sections are squares...
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Nine Definite Integrals and Fundamental Theorem Of Calculus Problems

For Students 11th - Higher Ed
For this definite integral worksheet, student find the value of five definite integrals.  Student then explain why a definite integral inequality holds true.  Finally, students are given the graph of a functions and asked to determine...
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A.P. Calculus Test Covering Limits, Derivatives, and Integrals

For Students 12th - Higher Ed
In this calculus worksheet, students solve ten multiple-choice questions and eleven free-response questions covering limits, derivatives, and integrals.  Students should have covered trig limits prior to this worksheet.
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Calculus Test

For Students 11th - Higher Ed
In this calculus activity, learners integrate various functions, find the sum of a series, and solve differential equations. There are 16 questions including multiple choice and free response.
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AP Calculus Practice Exam

For Students 11th - 12th
In this Calculus worksheet, students are provided with problems of a similar nature to those on their exam.  Problems include limits, derivatives, and integrals.  The four page worksheet contains twenty-eight problems.  Answers are not...
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Evaluate Integrals

For Students 12th - Higher Ed
In this evaluate integrals instructional activity, students solve and complete 12 different problems. First, they use the trigonometric identity given to rewrite the integral as follows. Then, students substitute when necessary and use...
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Evaluate Integrals

For Students 12th - Higher Ed
In this evaluating integrals worksheet, students solve and complete 12 various types of problems. First, they use the trigonometric identity to rewrite each integral. Then, students use integral formulas for sine function. They also...
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Computing Indefinite Integrals

For Teachers 11th - Higher Ed
Provide your students with a framework for introducing indefinite integrals. It takes into account a learner’s prior knowledge of slope fields and definite integrals and uses the notion of an antiderivative to give them an intuitive...
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2016 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
Using the common errors and test taker responses, it's easy for teachers to see where to pinpoint instruction throughout the AP Calculus course. Pupils, using the released items, develop an understanding of how future exams assess the...
5:41
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Integration by Parts

For Students 11th - Higher Ed
Class members view how to derive the formula to integrate by parts with a video that uses the Product Rule from the beginning of AP Calculus to derive the rule to integrate products. The resource shows the formula in two different forms,...
4:37
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Trigonometric Integrals

For Students 11th - Higher Ed
Present examples of trigonometric integrals with a video that shows an example of determining a trigonometric integral. The AP Calculus example shows that many of the trigonometric integrals require manipulation and using trigonometric...
10:30
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Integration by Partial Fractions

For Students 11th - Higher Ed
Demonstrate a complex integration by using partial fractions with an AP Calculus resource that works an example of integrating a rational function. The function's denominator has linear factors with one of them repeated. 
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Variational Calculus 1

For Students 12th - Higher Ed
In this methods of applied mathematics worksheet, students solve 4 different problems that relate to variational calculus. First, they determine a second order differential equation for the function y(x) for which the function given...
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Discovering a Fundamental Fact about Integration

For Teachers 11th - 12th
Sketch on coordinate axes to find the exact value of an integral shown. Individuals complete the data grid with the values of integrals returned and use inductive reasoning to guess the formula.
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Calculus Volume 3

For Students 11th - Higher Ed
See calculus to the end. The eBook is the last of a three-volume series covering college-level calculus topics. Scholars begin with parametric equations and polar coordinates before moving to vectors, finally ending with multi-variable...
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AP Calculus Exam Prep

For Teachers 11th - 12th Standards
Pupils practice sample questions from the AP Calculus Exam. They solve calculus problems involving bounded regions, integration, derivatives, and limits. They then answer free response and multiple choice questions on the above mentioned...
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The Second Fundamental Theorem of Calculus

For Teachers 10th - 12th Standards
Students investigate the fundamental theorem of calculus. In this calculus instructional activity, students derive the fundamental theorem of calculus. They differentiate between the first and second theorem.
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The First Fundamental Theorem of Calculus

For Teachers 9th - 12th Standards
Solve problems using the fundamental theorem. In this calculus lesson, learners solve problems using theorems and proving theorems. They derive the fundamental theorem as they study it.
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Integration By Substitution

For Teachers 9th - 12th Standards
Young scholars calculate the solution using integrals in this calculus instructional activity. They use the TI to create a visual of how to compute integrals. Learners also use substitution by integration and all its properties.
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Integration By Parts

For Teachers 9th - 12th Standards
Pupils solve problems using integration by parts in this calculus lesson. Learners apply the product rule and integration by parts. They graph the equation and use the TI to observe the integration process.
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Piecewise Linear Integral

For Teachers 9th - 12th Standards
Young mathematicians solve and graph piecewise functions in this calculus lesson. They integrate linear functions containing restrictions and graph their answers using the TI.
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Elevator Height as Integral of Velocity

For Teachers 9th - 12th Standards
Pupils investigate integrals and their relationship to velocity in this calculus lesson. They use the idea of a moving elevator to explore vertical motion and use the TI to help them create a visual.
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AP Calculus Practice Exam

For Students 11th - 12th
In this practice exam learning exercise, students solve 17 multiple choice problems. Students find derivatives, points of continuity, maximums, minimums, integrals, and area of enclosed regions.
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The Fundamental Theorem of Calculus

For Teachers 10th - 12th
Knowledge of definite integrals is fundamental to the understanding of calculus. Instructors of an AP® Calculus course consult a comprehensive guide to the Fundamental Theorem of Calculus. The document provides teaching suggestions and...
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Fundamental Theorem of Calculus

For Teachers 10th - 12th
Does it make more sense to teach the Second Fundamental Theorem of Calculus before the First Fundamental Theorem of Calculus? An AP® Calculus curriculum module explores the interesting question. A set of five worksheets first looks at...
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2018 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
Pupils interact with six released questions from the 2018 AP Calculus AB exam. Teachers use the additional materials to determine where the class is in terms of readiness for the exam to help provide additional instruction as needed.
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2018 AP® Calculus BC Free-Response Questions

For Students 10th - 12th
Don't just rely on the textbook. Released items from the AP Calculus BC provide pupils and teachers insight on how topics appear on the exam. The six items are divided into calculator and non-calculator sets with half of them also...
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2017 AP® Calculus BC Free-Response Questions

For Students 10th - 12th
Pupils use the six released free-response questions from an AP Calculus BC exam to prepare for their own exam. The questions cover a range of topics from the BC curriculum, with half of the items coming strictly from AB material and the...
2:29
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STEMbite: The Calculus of Building Blocks

For Students 9th - 12th
Watch this video with your junior engineers or your higher-level mathematicians. In it, Drew builds half of an arch bridge with building blocks. There happens to be a magnetic board behind the blocks, on which he affixes a magnet for...
18:26
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Limits | Essence of Calculus, Chapter 7

For Students 11th - Higher Ed
There's no limit on the importance of the limit in calculus. Scholars learn the formal definition of derivative and the epsilon-delta definition of limit by watching a video. They also listen to an explanation of l'Hospital's Rule.
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Volume by Disks: The Vase Case

For Students 11th - Higher Ed
Finding the volume is an integral characteristic of a vase. Using the idea that summing the areas of cross-sectional disks will calculate the volume of a rotational solid, pupils find the volume of a vase. Scholars determine the interval...
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Integral Worksheet

For Students Higher Ed
In this college level calculus worksheet, students evaluate the given definite integrals.  The three page worksheet contains eight problems.  Solutions are not provided. 
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Evaluating Definite Integrals With Substitution

For Teachers 9th - 12th
Students investigate how to evaluate the definite integral with using substitution. They evaluate expressions when using the substitution method to find the definite integral. The lesson plan includes example problems for the teacher to...
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Passive Integrator and Differentiator Circuits

For Students 10th - Higher Ed
In this circuits worksheet, students answer 25 questions about passive integrator circuits and passive differentiator circuits given schematics showing voltage. Students use calculus to solve the problems.
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2004 AP® Calculus AB Free-Response Questions Form B

For Students 10th - 12th
The test is not all about skills. The six free-response questions contain two real-world application problems. Those items deal with populations and velocities. The other four questions ask pupils to show their knowledge of calculus...
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2010 AP® Calculus BC Free-Response Questions

For Students 10th - 12th
What do a zoo, snow, and a moving particle have in common? The six free-response items from the 2010 AP® Calculus BC exam allow teachers and pupils see how items may show up in context. Three of the questions are from only the BC portion...
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Hot Tub

For Teachers 11th - 12th
Students investigate integral calculus.  In this calculus lesson, students explore an application of integrations through a leaking hot tub problem.  The activity emphasizes using the integral of a rate of change to give the accumulated...
4:26
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Average Value of a Function

For Students 11th - Higher Ed
Viewers learn how to find the average of a variable density with a video made for AP Calculus. The presenter applies the formula to find the average value of a function with an example that provides a density function for a rod. The...
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AP Calculus Practice Exam AB Version-Section1-Part 1

For Students 11th - Higher Ed
In this calculus practice exam, students select the best answer to 28 multiple choice questions. Questions cover information from the entire year of calculus. A suggested time limit of 55 minutes is included.
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Seventeen Various Integral Problems

For Students 11th - Higher Ed
In this integrals activity, students solve seventeen integral problems.  One problem covers completing the square, one curve length, one partial fraction decomposition, one Simpson's rule, and several are trigonometric integrals.
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Integral Expression

For Students 12th - Higher Ed
For this integral expression worksheet, students solve and complete 4 different parts of a problem. First, they write an integral expression in terms of average. Then, students find and indicate the units of measure in terms of temperature.
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Integrals - U Substitution (Part 1)

For Students 11th - 12th
Want to create more independent learning environment? The provided video gives a clear explanation of integration using U-substitution. Use is as a teaching tool or as supplemental material for your unit on limits.
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The Region Between Two Curves

For Teachers 11th - 12th
Students find the area between two curves.  In this area between two curves lesson, students find the area between the sine and cosine functions.  Students find the volume of the region when it is rotated around the x-axis.  Students...
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2012 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
What's on the test? Released items provide insight into the AP Calculus test. The six items are split into two groups, two calculator items and four non-calculator items. Resources allow teachers to see what kinds of answers test takers...
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2011 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
So this is the real test? The resource provides the free-response questions from the 2011 AP Calculus AB exam. The six questions are split into two sections, calculator and non-calculator items. Several resources give teachers...
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2013 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
What is the important stuff? The free-response questions from the 2013 Calculus AB exam highlight the topics in calculus assessed that year. Items provide pupils the opportunity to experience questions similar to what they will see on...
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2007 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
Grow your knowledge of Calculus AB. Items released from the 2007 AP® Calculus AB exam provide information to teachers about the importance of particular topics. Pupils use the questions to gain an understanding of ways the exam assesses...
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2014 AP® Calculus BC Free-Response Questions

For Students 10th - 12th
How much of the material is really important? Pupils use the free-response questions from the 2014 AP Calculus BC exam to see how the questions appear. With the supporting material, teachers understand the topics that appear on the test...
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2015 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
How will the topics be assessed? Using the six released items from the Calculus AB test, pupils see how the exam assesses content from the course. With the additional materials covering test taker's responses and scoring keys, teachers...
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2014 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
Give classes a sneak peek at an upcoming test. Scholars work released items from the AP Calculus AB test to prepare for their own exams. The resource consists of six items, two graphing calculator items, and four non-calculator items....
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2016 AP® Calculus BC Free-Response Questions

For Students 10th - 12th
The split is 50-50. Released free-response questions from the 2016 AP Calculus BC show that half the questions are from the AB portion of the course and the others are from the BC portion. Teachers and pupils use the items to prepare and...
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2009 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
What was one of the big takeaways from 2009? By reviewing the released AP® Calculus free-response questions, pupils and teachers realize that performing mathematics is not all there is to doing well on the exam. Several items require...
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2015 AP® Calculus BC Free-Response Questions

For Students 10th - 12th
How will the exam ask this question? The free-response questions from the 2015 AP Calculus BC exam demonstrate how questions appear on the test. Using the questions, pupils gain experience answering similar inquiries to what they see on...
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2006 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
Develop more than just a topical approach to the exam. The six free-response questions from 2006 reveal how the AP exam assesses topics. By using the questions, pupils practice topics such as velocity, sketch slope fields, solving...
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2010 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
Is this going to be on the test? The resource contains the AP Calculus AB free-response questions that were on the 2010 test. Pupils use the six questions to prepare for their tests. Teachers use the included response resources to...
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2013 AP® Calculus BC Free-Response Questions

For Students 10th - 12th
Apply the skills to problems to ace the AP test. Free-response questions require scholars to take calculus skills and apply them to situations. Pupils use the questions to see how those situations appear on an upcoming exam. Teacher use...
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2009 AP® Calculus BC Free-Response Questions

For Students 10th - 12th
Dive into the exam review. Pupils use questions, including one about diving into a pool, to prepare for the AP® Calculus BC exam. Items include contextual and non-contextual questions with contextual examples requiring a calculator....
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Improper Integrals

For Teachers 12th
Twelfth graders investigate the limitations of the Fundamental Theorem of Calculus.  In this calculus lesson plan, 12th graders explore when one can and cannot use the Fundamental Theorem of Calculus and explore the definition of an...
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Sum of Rectangle Areas to Approximate Integrals

For Teachers 11th - 12th
Young scholars explore the concept of approximating integrals.  In this approximating integrals lesson, students approximate integrals by finding the area of rectangles.  Young scholars use a graphing calculator to find the left, right,...
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Volume by Cross Sections

For Teachers 11th - 12th
Eleventh and twelfth graders find the volume of figures using cross sections. They use their Ti-Nspire to find the volume of a solid formed by cross sections of a function. Pupils find the integrals of the shapes using cross sections. 
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Calculus Volume 2

For Students 11th - Higher Ed
Delve into the middle of calculus. The resource is the second eBook of three that covers the complete calculus content. Seven chapters begin with integrals and different methods of integration before moving on to differential equations....
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Integration

For Students 12th - Higher Ed
In this calculus learning exercise, students simplify integrals and solve problems that involve related rates and maximum and minimums.  The one page learning exercise contains twenty-three problems.  Solutions are not provided. 
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Air Resistance

For Teachers 12th
Twelfth graders explore integral calculus.  For this Calculus lesson, 12th graders use the tools of integral calculus to investigate the motion of a free falling objet which is governed by both gravity and air resistance. 
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Variational Calculus 3

For Students 12th - Higher Ed
In this methods of applied mathematics worksheet, students solve 5 various types of problems that relate to variational calculus equations. First, they determine a second order differential equation for the geodesic. Then, students write...
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Variational Calculus 2

For Students 12th - Higher Ed
In this methods of applied mathematics worksheet, students solve 6 different types of problems related to variational calculus. First, they determine the first integrals of the Euler-Lagrange equations corresponding to the optimizations...
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Functions Defined by Integrals

For Teachers 10th - 12th
Functions defined by integrals need to be carefully integrated into the course. An AP® Calculus curriculum module discusses functions defined by integrals. It explains several reasons why pupils have difficulty with the concept and...
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AP® Calculus: Special Focus - Infinite Series

For Teachers 10th - 12th
Read a series to develop an understanding of series. The special focus professional development on infinite series consists of eight articles. Articles contain information strictly for teachers along with aspects to incorporate into the...
12:38
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What Does Area Have to Do with Slope? | Essence of Calculus, Chapter 9

For Students 11th - Higher Ed
It seems like area and slope wouldn't have any connection. Here is a video that explains how to find the average value of a function over an interval. It connects the area under a curve to the slope of the antiderivative curve.
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Computing Indefinite Integrals

For Teachers 9th - 12th
Students assess transformations to remove integral symbols as well as to simplify expressions. They explore the Symbolic Math Guide to assist them in solving indefinite integration by parts. This lesson includes partial fractions,...
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Exploring Taylor's Integrals

For Teachers 9th - 12th
Students are introduced to the use of Taylor's polynomials to integrate functions that are not easily integrated by hand. They complete a series of calculations that vary from integrating to finding a Taylor polynomial. Students...
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Integrals and Limits

For Students Higher Ed
In this college level calculus worksheet, students evaluate the given integrals and compute the given limits or show that the limit does not exist.  The three page worksheet contains ten problems.  Answers are not provided. 
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Integrals and Integration

For Students 11th - 12th
In this calculus worksheet, learners evaluate definite integrals using numbers and radians. They take the derivative to solve their equations, using trigonometric identities. There are 4 questions.
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Definite Integrals

For Students 11th - 12th
In this calculus worksheet, students calculate the definite integral given different functions. They evaluate the functions for given values. There are 20 questions.
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Integrals

For Students 12th - Higher Ed
In this integrals worksheet, students solve and complete 20 various types of problems. First, they evaluate each of the listed integrals shown. Then, students find the area under the graph given and above the x-axis between two points....
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Arc Length

For Teachers 11th - 12th
Students explore the concept of arc length.  In this arc length lesson, students find the arc length of a sine curve over a specific interval using integrals.  Students use their Ti-Nspire to determine the integrals of sine curves.
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Volumes by Washers

For Students 11th - Higher Ed
Here's an AP Calculus resource that works an example problem of finding the volume where the cross sections of a solid are washer shaped. In the example the axis of rotation is vertical, creating horizontal washers.
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Functions

For Students 12th - Higher Ed
In this functions instructional activity, students solve and complete 5 different sets of problems that include determining functions on intervals. First, they define a definite integral and the Fundamental Theorem of Calculus. Then,...
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Calculus Concepts

For Teachers 12th - Higher Ed
Young scholars use a graphing calculator to do calculations and edit their calculations using script. In this graphing calculator lesson, students compute the length of intervals given in parametric form. Young scholars verify the...
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I'm Lovin' It: Finding Area Between Curves

For Teachers 11th - Higher Ed Standards
Students explore the concept of finding the area between two curves. In this finding the area between two curves instructional activity, students model the logos of McDonalds, Nike, and Motorola on grid paper. Students find functions to...
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Shortest Path

For Teachers 9th - 12th
High schoolers explore an application of definite integrals. They graph three functions that go through the same three points. Learners use the symbolic capacity of their calculators to find the shortest path through the three points and...
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Cookie Recipe: Integrals

For Students 11th - Higher Ed
In this calculus worksheet, learners solve integrals over a defined interval in order to gain the measurements for the provided cookie recipe. There are 10 ingredients.
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Integral Test

For Teachers 12th
Twelfth graders investigate convergence of an infinite series.  In this Calculus instructional activity, 12th graders apply the integral test to determine convergence of an infinite series. 
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Area under curve

For Teachers 12th
Twelfth graders investigate an application of definite integrals.  In this Calculus lesson, 12th graders use the symbolic capacity of the TI-89 calculator and the concept of the definite integral to explore the area bounded by a function...
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Shortest Path

For Teachers 12th
Twelfth graders explore an application of the definite integrals.  In this calculus lesson, 12th graders graph three functions on the same domain and each goes through the same three points.  Students use the symbolic capacity of their...
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2003 AP® Calculus BC Free-Response Questions Form B

For Students 10th - 12th
Be integral to the learners in the classroom. Pupils use integrals, differentiation, and limits to solve the free-response questions from an AP® Calculus BC exam. Items cover topics such as area under a curve, particle motion, average...
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2005 AP® Calculus BC Free-Response Questions Form B

For Students 10th - 12th
Let's put it in context. Pupils use the AP® Calculus BC free-response questions to practice for the exam. Five of the six released items deal with mathematical problems, with one using a real-world situation. Learners use differentiation...
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2004 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
Develop a deeper understanding of calculus. The released item and support materials allows teachers and pupils to dig deeper into calculus topics. Scholars use the commentary and descriptions of the items to gain a better understanding...
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2009 AP® Calculus AB Free-Response Questions Form B

For Students 10th - 12th
Get a free peek into a calculus exam. Released free-response items from the AP® Calculus AB exam provide teachers and pupils examples of how topics may appear. The six questions cover topics such as growth rates, volumes of rotations,...
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2003 AP® Calculus AB Free-Response Questions Form B

For Students 10th - 12th
Get a better grasp of the intended content. The six free-response questions provide insight into the purpose of the AP® Calculus content. Two questions have real-world context while the remaining are void of context. The items range from...
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2005 AP® Calculus AB Free-Response Questions Form B

For Students 10th - 12th
Grab a peak inside the test. The 2005 free-response questions from the AP® Calculus exam allow learners to see how topics appear on the tests. Topics cover differential equations, bounded areas, particle motion, and slopes of tangent lines.
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2002 AP® Calculus AB Free-Response Questions Form B

For Students 10th - 12th
Become better at the test concepts. An educational resource presents six free-response questions from the AP® Calculus AB exam. Pupils use the items to practice the content, which contains concepts ranging from particle motion to ships...
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2008 AP® Calculus BC Free-Response Questions Form B

For Students 10th - 12th
This is integral to the test. Pupils realize that every question in the released free-response items from the 2008 AP® Calculus exam Form B requires integration. By reviewing the additional resources, scholars learn how important it is...
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2001 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
Take a closer look at the exam. Released items from the 2001 Calculus AB exam allow teachers and pupils to view how the exam presents questions. Scholars use the questions to prepare for the test and compare their responses to the...
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2003 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
Take a free look. Released free-response questions from the 2003 AP® Calculus AB exam show how questions cover different topics. Pupils and teachers review the topics of bounded regions, particle movement, rates, relationships between...
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2002 AP® Calculus BC Free-Response Questions

For Students 10th - 12th
Where do the questions come from? Individuals examine released items and see the topics included. Teachers use the materials to see the distribution between strictly BC content and that which is shared with AB. The 2002 Calculus BC is...
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2002 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
Peek behind the curtain. Using a helpful resource, teachers and pupils see the way concepts appear on the AP® Calculus exam. Scholars use the questions to prepare for their own exams, gaining quality practice before test day. Questions...
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2004 AP® Calculus BC Free-Response Questions

For Students 10th - 12th
Given a set of questions, pupils realize the way the Calculus BC free-response exam asks questions. The items include both contextual and non-contextual questions. Both types are split into the calculator and non-calculator sections for...
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2003 AP® Calculus BC Free-Response Questions

For Students 10th - 12th
There are shaded areas, particles, polar equations, oh my! Teachers use the materials to see how their pupils compare to the AP® Calculus BC exam expectations. Scholars use the released items to practice on several BC topics and become...
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2006 AP® Calculus AB Free-Response Questions Form B

For Students 10th - 12th
An exam divides the six free-response questions into calculator use and no calculator use items. The non-calculator items are conceptual in nature and test whether pupils understand the calculus concepts. Scholars explain their answers...
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2012 AP® Calculus BC Free-Response Questions

For Students 10th - 12th
Get better prepared. Individuals use the six released AP® Calculus BC free-response questions to practice for the actual exam. The questions contain two calculator items and four non-calculator items. Topics include particle movement,...
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2011 AP® Calculus BC Free-Response Questions

For Students 10th - 12th
Does the exam look pupils expected? Released free-response questions from the AP® Calculus BC exam allow teachers and pupils to see and practice with actual exam questions. Three of the questions come from the AB portion of the course...
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2007 AP® Calculus AB Free-Response Questions Form B

For Students 10th - 12th
What do the questions look like? Six free-response questions provide pupils with knowledge of how calculus topics appear on the AP® Calculus exam. The questions involve calculator and non-calculator problems by providing mathematical and...
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2010 AP® Calculus AB Free-Response Questions Form B

For Students 10th - 12th
Most say that practice makes perfect. Scholars use the released free-response questions to practice for the AP® Calculus AB exam. Each question comes with several parts and either do or do not allow calculators. Items include topics such...
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2009 AP® Calculus BC Free-Response Questions Form B

For Students 10th - 12th
Let them eat cake. The free-response questions from an AP® Calculus BC exam cover several topics that include finding the area under a curve within the context of creating a birthday cake. Calculus concepts include finding rates of...
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2002 AP® Calculus BC Free-Response Questions Form B

For Students 10th - 12th
Though the items may be older, they still provide information. Released free-response items from the 2002 AP® Calculus BC exam provide teachers and pupils with information about how topics appear on the exam. Questions come from the AB...
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2008 AP® Calculus AB Free-Response Questions Form B

For Students 10th - 12th
How would those items score? The resource provides pupils an opportunity to answer actual free-response questions and determine their scores. Items from the AP® Calculus AB exam from the 2008 administration demonstrate how topics appear...
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2004 AP® Calculus BC Free-Response Questions Form B

For Students 10th - 12th
Take another look at the exam. A set of questions is a different form for the 2004 AP® Calculus BC free-response question section. The questions are evenly split into calculator and non-calculator sections. Pupils use their knowledge of...
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1998 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
Take a peek into the exam. Released items from the Calculus AB exam provide pupils insight into how questions appear on the exam. The six questions range from finding the area under a curve and the volume of the revolution to working...
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1999 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
Gain an understanding of the test before taking your own. Scholars develop an understanding of how calculus concepts appear on the AP® exam. The six released questions come from across the curriculum and include particle movement and...
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2000 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
Gain an upper hand in preparation for the exam. Teachers and pupils use six released questions to gain an understanding of the AP® Calculus exam. By answering the questions, scholars see how different concept of the course show up on the...
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1998 AP® Calculus BC Free-Response Questions

For Students 10th - 12th
Provide a test run. Pupils use the questions to practice taking the free-response section of the AP® Calculus BC exam. Teachers see the different ways questions appear on the exam to prepare individuals for whatever comes their way on...
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1999 AP® Calculus BC Free-Response Questions

For Students 10th - 12th
Get a 20/20 view of the exam. Free-response questions from the 1999 Calculus BC exam provide an insight to the exams. Pupils review the questions to see how topics show up on the test. Topics range from particle motion along a function,...
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2000 AP® Calculus BC Free-Response Questions

For Students 10th - 12th
How are concepts divided? Pupils use the AP® Calculus free-response questions to see how the exam addresses concepts. The concepts are divided into those that use calculators and others that do not. Calculator items require finding...
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2011 AP® Calculus AB Free-Response Questions Form B

For Students 10th - 12th
Half are real-world. The 2011 AP® Calculus AB free-response questions let pupils and teachers see how content appears on the exam. Half of the questions contain real-world context, and four items do not allow calculators. Real-world...
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2011 AP® Calculus BC Free-Response Questions Form B

For Students 10th - 12th
Take a second look. Form B of the AP® Calculus BC free-response questions allow pupils and teachers to see another set of items. The six questions expand the set of items from the 2011 free-response questions. Form B items include a...
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2010 AP® Calculus BC Free-Response Questions Form B

For Students 10th - 12th
Keep moving along a curve. Two items in the set of released free-response questions from the 2010 AP® Calculus BC exam involve movement along a graph. One involves particle motion along a polar curve while the other uses a squirrel...
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2007 AP® Calculus BC Free-Response Questions Form B

For Students 10th - 12th
There is just a single real-world problem. Released free-response items from the 2007 AP® Calculus BC Form B contains only one real-world question. The question involves rate of change of wind chill scenario. The mathematical problems...
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Integral Exchange: Applying Integration By Substitution

For Students 11th - Higher Ed
In this math worksheet, students practice using the method of substitution with the integrals. They read the examples and then work in groups to apply the method.
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Engineering in a Calculus Classroom

For Teachers 11th - 12th
Learners solve several differential equations. In this calculus lesson, students test their modulus calculation by testing different weights or beam lengths. They share their findings in class.
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Video On Integration And Quiz On Induction

For Teachers 10th - 12th
Students engage in a lesson that is concerned with the concepts of integration and induction. They view a video about integration and take a quiz on induction. Students use the resource links to view information added to the lesson to...
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Fundamental Theorem of Calculus

For Teachers 9th - 12th Standards
High schoolers, with the assistance of their TI-84 Plus / TI-83 Plus calculators, explore and assess the connections between an accumulation function, one defined by a definite integral, and the integrand. They summarize that the...
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Filling The Urn

For Teachers 9th - 12th Standards
Students calculate the area under a curve. In this calculus instructional activity,students calculate the area by integrating the equation. They calculate the volume of cylindrical shapes created between graphs.
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Numerical Word Search

For Students 12th - Higher Ed
In this numerical word search learning exercise, students solve ten problems and then search for the solutions in a grid of numbers.  A few of the problems require that the student has taken calculus. 
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Integration Crossing

For Students 11th - 12th
In this Calculus learning exercise, students use the TI-89 calculator to compute derivatives and anti-derivatives in order to solve a crossword puzzle. The two page learning exercise contains twenty-one problems. Answers are not provided. 
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Functioning on Your Own

For Teachers 12th - Higher Ed
Students explore and create functions. In this calculus instructional activity, students create functions with the help of the Voyage 200 PLT. This assignment requires technology.
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Little Red Corvette

For Teachers 11th - 12th
Students investigate the acceleration of a car.  In this acceleration of a car lesson, students make a scatter plot of corvette acceleration data.  Students find a model to fit the data by taking the integral of the curve created by the...
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Work and Force: Lifting a Bucket

For Students 11th - Higher Ed
How much work does it take to lift a bucket? An interactive presents a problem of lifting a bucket from the ground to the top of a building. Using their knowledge about work and integrals, pupils calculate the amount of work required to...
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Volume by Cross Section: Volume of the Cone

For Students 11th - Higher Ed
Discover another way to find the volume of a cone. Pupils explore how the area of a cross section changes as it moves through a cone. The interactive uses that knowledge to develop the integral to use to find the volume of the cone....
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Method of Cylindrical Shells

For Students 11th - Higher Ed
Approximate the volume of a solid of revolution. Using a method similar to approximating the area under a curve, pupils investigate the volume of a solid of revolution. The learners use a given definite integral to find the volume of...
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Approximating Integrals with Riemann Sums

For Teachers 9th - 12th
Students practice calculating and analyzing Riemann sums and illustrate when Riemann sums will over/under-approximate a definite integral. They view how the convergence of Riemann sums as the number of subintervals get larger.
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Categorize Twelve Integrals

For Students 11th - Higher Ed
In this integral worksheet, students categorize the given twelve integrals to determine which of the four basic integral functions provided each follows.
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Exploration 4: Definite Integrals by Trapezoidal Rule

For Students 10th - 12th
In this integral instructional activity, students geometrically estimate distance in given problems. They use the trapezoidal rule to determine definite integrals. Students explore distance and velocity. This three-page instructional...
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Integration By Parts

For Teachers 11th - 12th
This lesson plan provides an introduction to integration by parts. It helps learners first recognize derivatives produced by the product rule and then continues with step-by-step instructions on computing these integrals. It also shows...
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The Integration of Exponential Functions

For Students 11th - 12th
For this calculus worksheet, students solve problems using integration. They solve exponential functions using differentiation formulas. There are 12 problems with an answer key.
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The Method of Integration by Parts

For Students 11th - 12th
In this calculus worksheet, students perform integration by parts. They solve differential equations as they use integration by part to solve unlike terms. There are 23 problems with an answer key.
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The Method of Integration by Partial Fractions

For Students 9th - 10th
In this calculus worksheet, students solve problems using integration by partial fractions. They add fractions to get a common denominator, then take the derivative. There are 20 questions with an answer key.
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The Limit Definition of a Definite Integral

For Students 11th - 12th
In this calculus worksheet, students calculate the derivative and integral of different functions. They use interval to mark the beginning and end of their calculations. There are 15 questions with an answer key.
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Solving Differential Equations via the Fundamental Theorem of Calculus

For Teachers 11th - 12th
Students use their TI-89 calculator to compute derivatives and anti-derivatives. In this differential equation lesson, students follow detailed directions to complete one table of derivative/anti-derivative values. They compute the...
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What is an Antiderivative? Slope Fields

For Teachers 10th - 12th
Students take the derivative of a function. In this calculus lesson, students define slope fields and what they are used for. They relate slope fields to the real world.
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Worksheet 28 - Fall 1995

For Students 11th - Higher Ed
For this math worksheet, learners define a definite integral. Then they draw a picture or sequence of pictures with progressively finer partitions..
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Math 155 Review Worksheet 10

For Students Higher Ed
In this college level Calculus activity, students evaluate indefinite and definite integrals.  The two page activity contains eight problems.  Answers are not included. 
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Eight Integrals Using Partial Fractions

For Students 12th - Higher Ed
In this integral worksheet, students integrate eight problems using partial fraction decomposition.  A review on partial fractions is included at the top of the worksheet.
10:01
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Derivatives

For Students 9th - 12th Standards
Physics and math are closely associated in the science community. High schoolers use information in the video to determine how derivatives and calculus help them understand our local environment. They complete problems that relate to...
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Calculus Exam

For Students 11th
In For this calculus worksheet, student solve 6 multi-step problems involving volume, intervals, acceleration, and graphing. A graphing calculator will be necessary to complete the worksheet.
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Polar Functions

For Teachers 9th - 12th
Explore the concept of polar functions and learn how to determine the length around a function using polar coordinates. High schoolers use their Ti-calculators to determine the length of a function that looks like a five-leaved rose....
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Multivariable Calculus Quiz 9

For Students 11th - Higher Ed
In this math worksheet, students practice the calculus problems that involve integrals. Then they use the rules to complete the quiz by showing all the work for the two problems.
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Fun Times with Integrals

For Teachers 11th - 12th
Students study techniques used for solving integrals. Using tangrams and hands-on activities, students perform basic integration techniques. They use artistic skills and real-world applications to discover more complex methods of...
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Twenty Various Multiple Choice and Free-response Indefinite Integrals

For Students 11th - Higher Ed
In this integrals worksheet, students integrate various indefinite integrals in six multiple choice and fourteen  free response problems.  These problems include: trigonometric functions, u substitutions, and Ln.
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Arc Length

For Teachers 11th - 12th
Students determine the length of the sine curve on the interval 0 to 2Pi.  In this length of the sine curve on the interval 0 to 2Pi lesson, students use triangles and rectangles to estimate the arc length of the sine curve.  Students...
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The Area Function

For Teachers 11th - 12th
Students explore the concept of area under a curve.  In this area under a curve lesson plan, students find integrals of various functions.  Students use their Ti-Nspire to graph functions and find the area under the curve using the...
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Riemann Sums

For Teachers 11th - 12th
Students explore Riemann Sums.  In this Calculus lesson, students investigate Riemann Sums as an approximation of the value of the definite integral.
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Quartic Regions

For Teachers 11th - 12th
Students explore the concept of quartic regions.  In this quartic regions instructional activity, students find the area under a curve using their Ti-89 calculator.  Students find definite integrals of two curves.  Students determine the...
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Challenge Sheet 2: Evaluate Functions

For Students Higher Ed
In this function activity, students evaluate integrals using the fundamental theorem of calculus. They identify the maxima and minima on intervals. This one-page activity contains seven problems.
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AP Calculus Practice Exam

For Students 11th - 12th
In this Calculus worksheet, learners are provided with practice problems for their exam.  Topics covered include derivatives, area bounded by a curve, local maximum, instantaneous rate of change, and the volume of a solid of revolution. ...
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Simpson's Rule

For Students Higher Ed
In this Simpson's Rule worksheet, students use the Fundamental Theorem of Calculus to determine the antiderivatives and evaluate integrals. They find the midpoint of given polygons and sketch the graph of the function. This nine-page...
8:48
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Volumes by Slicing

For Students 11th - Higher Ed
Find the volume of a rotational solid by integrating the area of a slice. The AP Calculus resource works through an example of finding the volume of a solid created by rotating a function about the x-axis. The presentation breaks the...
16:11
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Finding The Area Under a Curve Using Rectangles

For Students 11th - Higher Ed
Here's an AP Calculus video that provides a discussion on determining an approximate for the area under a curve by calculating the sum of areas of rectangles that is smaller than the area and by finding the sum of areas of rectangles...
5:17
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Separation of Variables and Initial Value Problem

For Students 11th - Higher Ed
Ap Calculus classes learn how to solve a differential equation by getting like variables together with a video that presents an example of how to solve a differential equation by using the method of separation of variables. The example...
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Twenty-Eight Multiple Choice AP Calculus BC Exam Preparation Questions

For Students 11th - Higher Ed
In this AP Calculus review worksheet, students solve twenty-eight multiple choice questions in preparation for the AP Calculus BC exam.
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Math 224 Multivariable Calculus/ Bonus Quiz 9A

For Students 11th - Higher Ed
In this math worksheet, students practice solving the problems that involve integrals. They take the quiz applying the rules and showing all the work for the two problems.

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