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Joseph W. Cutrone, PhD
This course continues your study of calculus by introducing the notions of series, sequences, and integration. These foundational tools allow us to develop the theory and applications of the second major tool of calculus: the integral. Rather than measure rates of change, the integral provides a means for measuring the accumulation of a quantity over some interval of input values. This notion of accumulation can be applied to different quantities, including money, populations, weight, area, volume, and air pollutants. The concepts in this course apply to many other disciplines outside of traditional mathematics. Through projects, we will apply the tools of this course to analyze and model real world data, and from that analysis give critiques of policy.
Read more
This course continues your study of calculus by introducing the notions of series, sequences, and integration. These foundational tools allow us to develop the theory and applications of the second major tool of calculus: the integral. Rather than measure rates of change, the integral provides a means for measuring the accumulation of a quantity over some interval of input values. This notion of accumulation can be applied to different quantities, including money, populations, weight, area, volume, and air pollutants. The concepts in this course apply to many other disciplines outside of traditional mathematics. Through projects, we will apply the tools of this course to analyze and model real world data, and from that analysis give critiques of policy.
Read more
This course continues your study of calculus by introducing the notions of series, sequences, and integration. These foundational tools allow us to develop the theory and applications of the second major tool of calculus: the integral. Rather than measure rates of change, the integral provides a means for measuring the accumulation of a quantity over some interval of input values. This notion of accumulation can be applied to different quantities, including money, populations, weight, area, volume, and air pollutants. The concepts in this course apply to many other disciplines outside of traditional mathematics. Through projects, we will apply the tools of this course to analyze and model real world data, and from that analysis give critiques of policy.
Following the pattern as with derivatives, several important methods for calculating accumulation are developed. Our course begins with the study of the deep and significant result of the Fundamental Theorem of Calculus, which develops the relationship between the operations of differentiation and integration. If you are interested in learning more advanced mathematics, this course is the right course for you.
Register for this course and see more details by visiting:
OpenCourser.com/course/eo3217/calculus
What's inside
Syllabus
Module 1: Sequences and Series
Calculus is divided into two halves: differentiation and integration. In this module, we introduce the process of integration. First we will see how the definite integral can be used to find the area under the graph of a curve. Then, we will investigate how differentiation and integration are inverses of each other, through the Fundamental Theorem of Calculus. Finally, we will learn about the indefinite integral, and use some strategies for computing integrals.
Read more
Syllabus
Module 1: Sequences and Series
Calculus is divided into two halves: differentiation and integration. In this module, we introduce the process of integration. First we will see how the definite integral can be used to find the area under the graph of a curve. Then, we will investigate how differentiation and integration are inverses of each other, through the Fundamental Theorem of Calculus. Finally, we will learn about the indefinite integral, and use some strategies for computing integrals.
Read more
Module 2: The Definite Integral
In this module, we introduce the notion of Riemann Sums. In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum, named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lines on a graph, but also the length of curves and other approximations. This notion of approximating the accumulation of area under a group will lead to the concept of the definite integral, and the many applications that follow.
Module 3: The Fundamental Theorem of Calculus
We now introduce the first major tool of our studies, the Fundamental Theorem of Calculus. This deep theorem links the concept of differentiating a function with the concept of integrating a function. The theorem will consists of two parts, the first of which implies the existence of antiderivatives for continuous functions and the second of which plays a larger role in practical applications. The beauty and practicality of this theorem allows us to avoid numerical integration to compute integrals, thus providing a better numerical accuracy.
Module 4: The Indefinite Integral
In this module, we focus on developing our ability to find antiderivatives, or more generally, families of antiderivatives. In calculus, the general family of antiderivatives is denoted with an indefinite integral, and the process of solving for antiderivatives is called antidifferentiation. This is the opposite of differentiation and completes our knowledge of the two major tools of calculus.
Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over an interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.
Integration with Calculators and Tables
While the technique of finding antiderivatives is useful, there are some functions that are just too difficult to find antiderivatives for. In cases like these, we want to have a numerical method to approximate the definite integral. In this module, we introduce two techniques for solving complicated integrals: using technology or tables of integrals, as well as estimation techniques. We then apply our knowledge to analyze strategies and decision theory as applied to random events.
Good to know
Good to know
Know what's good ,
what to watch for , and
possible dealbreakers
Explores sequences, series, and integration, which are foundational calculus concepts
Provides a link between differentiation and integration through the Fundamental Theorem of Calculus
Develops the theory and applications of integration, a key tool for measuring accumulation and modeling real-world data
Taught by Joseph W. Cutrone, PhD, an experienced instructor in calculus and its applications
Introduces the concepts of accumulation and Riemann Sums, leading to the development of the definite integral
Offers practical methods for calculating integrals, including the use of technology and tables of integrals
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Reviews summary
Calculus thru data & modelling: series integration
Learners say this great course with engaging assignments provides a straightforward approach to integrals which serves as a good refreshment for previous engineering topics.
Activities
Be better prepared
before
your course. Deepen your understanding
during
and
after
it. Supplement your coursework and achieve mastery of the topics covered
in Calculus through Data & Modelling: Series and Integration with these
activities:
Organize Calculus Class Materials
Show steps
Establish a system for organizing your notes, assignments, and study materials to enhance your ability to review and retain information.
Browse courses on
Note-Taking
Show steps
-
Create a designated space for storing calculus-related materials.
-
Use a binder, notebook, or digital tool to organize your notes by topic.
-
Keep track of all assignments and quizzes.
-
Review your organized materials regularly to reinforce your learning.
Review Algebra and Trigonometry
Show steps
Ensure you have a strong foundation in algebra and trigonometry, which are essential prerequisites for calculus.
Browse courses on
Algebra
Show steps
-
Review basic algebraic operations, such as solving equations and manipulating polynomials.
-
Recall trigonometric identities and functions.
-
Practice solving algebraic and trigonometric problems.
Read 'Calculus: Early Transcendentals' by Stewart
Show steps
Strengthen your understanding of the fundamentals of integration by reviewing a comprehensive textbook.
View
Algebra and Trigonometry
on Amazon
Show steps
-
Read through the chapters covering integration.
-
Take notes and highlight important concepts.
-
Work through the practice problems at the end of each chapter.
Five other activities
Expand to see all activities and additional details
Show all eight activities
Explore Online Tutorials on Integration
Show steps
Supplement your learning by seeking out online tutorials that provide additional explanations and examples of integration techniques.
Show steps
-
Search for reputable online resources that offer tutorials on integration.
-
Follow along with the tutorials, taking notes and working through the examples.
-
Apply what you have learned to solve practice problems.
Participate in Integration Study Groups
Show steps
Collaborate with peers to reinforce your understanding of integration and enhance your problem-solving skills.
Browse courses on
Integration
Show steps
-
Identify interested classmates and form a study group.
-
Meet regularly to discuss integration concepts, solve problems together, and support each other's learning.
-
Engage in active listening, asking questions, and providing feedback.
Practice Integration Techniques
Show steps
Deepen your understanding of integration techniques by solving a variety of practice problems.
Browse courses on
Indefinite Integrals
Show steps
-
Find the indefinite integral of a given function.
-
Evaluate the definite integral of a function over a given interval.
-
Apply integration techniques such as substitution, integration by parts, and partial fractions.
Create a Visual Guide to Integration
Show steps
Enhance your understanding and ability to explain integration by creating a visual guide that illustrates key concepts.
Browse courses on
Fundamental Theorem of Calculus
Show steps
-
Identify the main concepts and steps involved in integration.
-
Choose appropriate visuals and examples to illustrate these concepts.
-
Create a visually engaging and informative guide.
Develop a Calculus Project Involving Integration
Show steps
Enhance your understanding and application of integration by working on a project that involves modeling a real-world problem.
Browse courses on
Calculus
Show steps
-
Identify a problem or phenomenon that can be modeled using integration.
-
Develop a mathematical model that incorporates integration.
-
Collect and analyze data to validate the model.
-
Present your findings in a report or presentation.
Organize Calculus Class Materials
Show steps
Establish a system for organizing your notes, assignments, and study materials to enhance your ability to review and retain information.
Browse courses on
Note-Taking
Show steps
Show steps
Show steps
- Create a designated space for storing calculus-related materials.
- Use a binder, notebook, or digital tool to organize your notes by topic.
- Keep track of all assignments and quizzes.
- Review your organized materials regularly to reinforce your learning.
Review Algebra and Trigonometry
Show steps
Ensure you have a strong foundation in algebra and trigonometry, which are essential prerequisites for calculus.
Browse courses on
Algebra
Show steps
Show steps
Show steps
- Review basic algebraic operations, such as solving equations and manipulating polynomials.
- Recall trigonometric identities and functions.
- Practice solving algebraic and trigonometric problems.
Read 'Calculus: Early Transcendentals' by Stewart
Show steps
Strengthen your understanding of the fundamentals of integration by reviewing a comprehensive textbook.
View
Algebra and Trigonometry
on Amazon
Show steps
Show steps
Show steps
- Read through the chapters covering integration.
- Take notes and highlight important concepts.
- Work through the practice problems at the end of each chapter.
Five other activities
Expand to see all activities and additional details
Show all eight activities
Explore Online Tutorials on Integration
Show steps
Supplement your learning by seeking out online tutorials that provide additional explanations and examples of integration techniques.
Show steps
Show steps
Show steps
- Search for reputable online resources that offer tutorials on integration.
- Follow along with the tutorials, taking notes and working through the examples.
- Apply what you have learned to solve practice problems.
Participate in Integration Study Groups
Show steps
Collaborate with peers to reinforce your understanding of integration and enhance your problem-solving skills.
Browse courses on
Integration
Show steps
Show steps
Show steps
- Identify interested classmates and form a study group.
- Meet regularly to discuss integration concepts, solve problems together, and support each other's learning.
- Engage in active listening, asking questions, and providing feedback.
Practice Integration Techniques
Show steps
Deepen your understanding of integration techniques by solving a variety of practice problems.
Browse courses on
Indefinite Integrals
Show steps
Show steps
Show steps
- Find the indefinite integral of a given function.
- Evaluate the definite integral of a function over a given interval.
- Apply integration techniques such as substitution, integration by parts, and partial fractions.
Create a Visual Guide to Integration
Show steps
Enhance your understanding and ability to explain integration by creating a visual guide that illustrates key concepts.
Browse courses on
Fundamental Theorem of Calculus
Show steps
Show steps
Show steps
- Identify the main concepts and steps involved in integration.
- Choose appropriate visuals and examples to illustrate these concepts.
- Create a visually engaging and informative guide.
Develop a Calculus Project Involving Integration
Show steps
Enhance your understanding and application of integration by working on a project that involves modeling a real-world problem.
Browse courses on
Calculus
Show steps
Show steps
Show steps
- Identify a problem or phenomenon that can be modeled using integration.
- Develop a mathematical model that incorporates integration.
- Collect and analyze data to validate the model.
- Present your findings in a report or presentation.
Career center
Learners who complete Calculus through Data & Modelling: Series and Integration will develop knowledge and skills
that may be useful to these careers:
Data Analyst
Data Analyst
Data analysts are responsible for collecting, cleaning, and analyzing data to help businesses make informed decisions. This course provides a solid foundation in the mathematical concepts used in data analysis, such as series, sequences, and integration. The ability to use these concepts to model and analyze data is a valuable skill for any data analyst.
Financial Analyst
Financial Analyst
Financial analysts use mathematical and statistical techniques to analyze financial data and make recommendations for investments. This course provides a strong foundation in the mathematical concepts used in financial analysis, such as series, sequences, and integration. The ability to use these concepts to model and analyze financial data is a valuable skill for any financial analyst.
Operations Research Analyst
Operations Research Analyst
Operations research analysts use mathematical and statistical techniques to analyze and improve business operations. This course provides a strong foundation in the mathematical concepts used in operations research, such as series, sequences, and integration. The ability to use these concepts to model and analyze business operations is a valuable skill for any operations research analyst.
Actuary
Actuary
Actuaries use mathematical and statistical techniques to assess risk and uncertainty. This course provides a strong foundation in the mathematical concepts used in actuarial science, such as series, sequences, and integration. The ability to use these concepts to model and analyze risk and uncertainty is a valuable skill for any actuary.
Statistician
Statistician
Statisticians use mathematical and statistical techniques to collect, analyze, and interpret data. This course provides a strong foundation in the mathematical concepts used in statistics, such as series, sequences, and integration. The ability to use these concepts to collect, analyze, and interpret data is a valuable skill for any statistician.
Quantitative Analyst
Quantitative Analyst
Quantitative analysts use mathematical and statistical techniques to analyze financial data and make recommendations for investments. This course provides a strong foundation in the mathematical concepts used in quantitative analysis, such as series, sequences, and integration. The ability to use these concepts to model and analyze financial data is a valuable skill for any quantitative analyst.
Software Engineer
Software Engineer
Software engineers design, develop, and maintain software systems. This course provides a strong foundation in the mathematical concepts used in software engineering, such as series, sequences, and integration. The ability to use these concepts to model and design software systems is a valuable skill for any software engineer.
Data Scientist
Data Scientist
Data scientists use mathematical and statistical techniques to collect, analyze, and interpret data. This course provides a strong foundation in the mathematical concepts used in data science, such as series, sequences, and integration. The ability to use these concepts to collect, analyze, and interpret data is a valuable skill for any data scientist.
Machine Learning Engineer
Machine Learning Engineer
Machine learning engineers design and develop machine learning models to solve real-world problems. This course provides a strong foundation in the mathematical concepts used in machine learning, such as series, sequences, and integration. The ability to use these concepts to design and develop machine learning models is a valuable skill for any machine learning engineer.
Business Analyst
Business Analyst
Business analysts use mathematical and statistical techniques to analyze business data and make recommendations for improvements. This course provides a strong foundation in the mathematical concepts used in business analysis, such as series, sequences, and integration. The ability to use these concepts to analyze business data is a valuable skill for any business analyst.
Economist
Economist
Economists use mathematical and statistical techniques to analyze economic data and make recommendations for economic policy. This course provides a strong foundation in the mathematical concepts used in economics, such as series, sequences, and integration. The ability to use these concepts to analyze economic data is a valuable skill for any economist.
Market Researcher
Market Researcher
Market researchers use mathematical and statistical techniques to collect and analyze data about consumers and markets. This course provides a strong foundation in the mathematical concepts used in market research, such as series, sequences, and integration. The ability to use these concepts to collect and analyze data about consumers and markets is a valuable skill for any market researcher.
Investment Analyst
Investment Analyst
Investment analysts use mathematical and statistical techniques to analyze financial data and make recommendations for investments. This course provides a strong foundation in the mathematical concepts used in investment analysis, such as series, sequences, and integration. The ability to use these concepts to analyze financial data is a valuable skill for any investment analyst.
Risk Analyst
Risk Analyst
Risk analysts use mathematical and statistical techniques to assess risk and uncertainty. This course provides a strong foundation in the mathematical concepts used in risk analysis, such as series, sequences, and integration. The ability to use these concepts to assess risk and uncertainty is a valuable skill for any risk analyst.
Teacher
Teacher
Teachers use mathematical and statistical techniques to teach students about mathematics and statistics. This course provides a strong foundation in the mathematical concepts that are taught in schools, such as series, sequences, and integration. The ability to use these concepts to teach students about mathematics and statistics is a valuable skill for any teacher.
For more career information including salaries, visit:
OpenCourser.com/course/eo3217/calculus
Reading list
We've selected seven books
that we think will supplement your
learning. Use these to
develop background knowledge, enrich your coursework, and gain a
deeper understanding of the topics covered in
Calculus through Data & Modelling: Series and Integration.
Provides a comprehensive overview of calculus, including the topics covered in this course. It valuable resource for students who want to supplement their learning or review the material in more depth.
Provides a comprehensive overview of calculus, including the topics covered in this course. It valuable resource for students who want to supplement their learning or review the material in more depth.
Save
Provides a comprehensive overview of calculus, including the topics covered in this course. It valuable resource for students who want to supplement their learning or review the material in more depth.
Provides a comprehensive overview of calculus, including the topics covered in this course. It valuable resource for students who want to supplement their learning or review the material in more depth.
Save
Provides a thorough introduction to calculus, including the topics covered in this course. It good choice for students who want a more rigorous treatment of the material.
Save
Provides an introduction to calculus specifically tailored for students in the life sciences. It covers the topics covered in this course, as well as additional topics that are relevant to biology, chemistry, and other life science disciplines.
Provides an introduction to calculus with a focus on applications in business, economics, and other social sciences. It covers the topics covered in this course, as well as additional topics that are relevant to these fields.
For more information about how these books relate to this course, visit:
OpenCourser.com/course/eo3217/calculus
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Level
Intermediate
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Institution
Johns Hopkins University
Instructor
Joseph W. Cutrone, PhD
Language
English
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This course belongs to a
collection
called
Integral Calculus through Data and Modeling. It's comprised of four courses:
- Calculus through Data & Modelling: Techniques of Integration
- Calculus through Data & Modelling: Vector Calculus
- Calculus through Data & Modelling: Series and Integration (viewing)
- Calculus through Data & Modelling: Integration Applications
Good to know
Explores sequences, series, and integration, which are foundational calculus concepts
Provides a link between differentiation and integration through the Fundamental Theorem of Calculus
Develops the theory and applications of integration, a key tool for measuring accumulation and modeling real-world data
Taught by Joseph W. Cutrone, PhD, an experienced instructor in calculus and its applications
Introduces the concepts of accumulation and Riemann Sums, leading to the development of the definite integral
Offers practical methods for calculating integrals, including the use of technology and tables of integrals
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