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Robert Ghrist
Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences. Distinguishing features of the course include: 1) the introduction and use of Taylor series and approximations from the beginning; 2) a novel synthesis of discrete and continuous forms of Calculus; 3) an emphasis on the conceptual over the computational; and 4) a clear, dynamic, unified approach.
Read more
Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences. Distinguishing features of the course include: 1) the introduction and use of Taylor series and approximations from the beginning; 2) a novel synthesis of discrete and continuous forms of Calculus; 3) an emphasis on the conceptual over the computational; and 4) a clear, dynamic, unified approach.
Read more
Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences. Distinguishing features of the course include: 1) the introduction and use of Taylor series and approximations from the beginning; 2) a novel synthesis of discrete and continuous forms of Calculus; 3) an emphasis on the conceptual over the computational; and 4) a clear, dynamic, unified approach.
In this third part--part three of five--we cover integrating differential equations, techniques of integration, the fundamental theorem of integral calculus, and difficult integrals.
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What's inside
Syllabus
Integrating Differential Equations
Our first look at integrals will be motivated by differential equations. Describing how things evolve over time leads naturally to anti-differentiation, and we'll see a new application for derivatives in the form of stability criteria for equilibrium solutions.
Read more
Syllabus
Integrating Differential Equations
Our first look at integrals will be motivated by differential equations. Describing how things evolve over time leads naturally to anti-differentiation, and we'll see a new application for derivatives in the form of stability criteria for equilibrium solutions.
Read more
Techniques of Integration
Since indefinite integrals are really anti-derivatives, it makes sense that the rules for integration are inverses of the rules for differentiation. Using this perspective, we will learn the most basic and important integration techniques.
The Fundamental Theorem of Integral Calculus
Indefinite integrals are just half the story: the other half concerns definite integrals, thought of as limits of sums. The all-important *FTIC* [Fundamental Theorem of Integral Calculus] provides a bridge between the definite and indefinite worlds, and permits the power of integration techniques to bear on applications of definite integrals.
Dealing with Difficult Integrals
The simple story we have presented is, well, simple. In the real world, integrals are not always so well-behaved. This last module will survey what things can go wrong and how to overcome these complications. Once again, we find the language of big-O to be an ever-present help in time of need.
Good to know
Good to know
Know what's good ,
what to watch for , and
possible dealbreakers
Develops essential Calculus skills, making it suitable for engineering, physical, and social science students
Introduces Taylor series and approximations at the outset, providing a robust mathematical foundation
Synthesizes discrete and continuous forms of Calculus, facilitating a comprehensive understanding
Emphasizes conceptual understanding over computational prowess, fostering a deep grasp of Calculus principles
Provides a clear, unified approach to Calculus, ensuring a coherent learning experience
The course is part of a five-part series, suggesting a comprehensive exploration of Calculus topics
Save this course
Save Calculus: Single Variable Part 3 - Integration to your list so you can find it easily later:
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Reviews summary
Calculus: part 3 - integration
Learners say that integration is well taught in this course with engaging assignments. Students say that while the course is difficult, it is well-received.
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: Single Variable Part 3 - Integration with these
activities:
Review Differentiation
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Differentiation is a prerequisite for integral calculus. This activity will help you to refresh your understanding of differentiation.
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Differentiation
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-
Review the definition of the derivative
-
Learn the basic techniques of differentiation
-
Practice differentiating a variety of functions
Review Single-Variable Calculus
Show steps
This activity will help you to refresh your knowledge of single-variable calculus. This knowledge will be a foundation for your study of integral calculus.
Browse courses on
Calculus
Show steps
-
Review your notes from previous calculus courses
-
Read a calculus textbook
-
Watch calculus videos online
Techniques of Integration Tutorial
Show steps
Integrals are the inverse of derivatives and are used to solve a wide range of problems in calculus. This tutorial will guide you through the basic techniques of integration, including substitution, integration by parts, and trigonometric integration.
Show steps
-
Review the definition of the integral
-
Learn the basic techniques of integration
-
Practice integrating a variety of functions
Five other activities
Expand to see all activities and additional details
Show all eight activities
Practice Integrals
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Practice is essential for mastering the techniques of integration. This activity provides a set of practice problems that will help you to improve your skills.
Show steps
-
Solve a variety of practice problems
-
Check your answers against the provided solutions
-
Review the problems you got wrong
Study Group
Show steps
Studying with peers can be a great way to improve your understanding of the material. This activity will help you to find a study group and to develop effective study habits.
Show steps
-
Find a study group
-
Meet with your study group regularly
-
Review the material together
-
Work on practice problems together
Calculus Workshop
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This activity will provide you with an opportunity to work with an expert calculus tutor and to receive personalized feedback on your work.
Browse courses on
Calculus
Show steps
-
Attend a calculus workshop
-
Work on practice problems with the tutor
-
Receive feedback on your work
Calculus Poster
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Creating a poster can be a great way to summarize and review the key concepts of calculus.
Browse courses on
Calculus
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-
Choose a topic for your poster
-
Research your topic
-
Create a visual representation of your topic
-
Write a brief explanation of your topic
Contribute to Calculus Open Source Project
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Contributing to an open source calculus project can be a great way to deepen your understanding of the subject and to give back to the community.
Browse courses on
Calculus
Show steps
-
Find an open source calculus project to contribute to
-
Read the project documentation
-
Make a contribution to the project
-
Review and test your contribution
Review Differentiation
Show steps
Differentiation is a prerequisite for integral calculus. This activity will help you to refresh your understanding of differentiation.
Browse courses on
Differentiation
Show steps
Show steps
Show steps
- Review the definition of the derivative
- Learn the basic techniques of differentiation
- Practice differentiating a variety of functions
Review Single-Variable Calculus
Show steps
This activity will help you to refresh your knowledge of single-variable calculus. This knowledge will be a foundation for your study of integral calculus.
Browse courses on
Calculus
Show steps
Show steps
Show steps
- Review your notes from previous calculus courses
- Read a calculus textbook
- Watch calculus videos online
Techniques of Integration Tutorial
Show steps
Integrals are the inverse of derivatives and are used to solve a wide range of problems in calculus. This tutorial will guide you through the basic techniques of integration, including substitution, integration by parts, and trigonometric integration.
Show steps
Show steps
Show steps
- Review the definition of the integral
- Learn the basic techniques of integration
- Practice integrating a variety of functions
Five other activities
Expand to see all activities and additional details
Show all eight activities
Practice Integrals
Show steps
Practice is essential for mastering the techniques of integration. This activity provides a set of practice problems that will help you to improve your skills.
Show steps
Show steps
Show steps
- Solve a variety of practice problems
- Check your answers against the provided solutions
- Review the problems you got wrong
Study Group
Show steps
Studying with peers can be a great way to improve your understanding of the material. This activity will help you to find a study group and to develop effective study habits.
Show steps
Show steps
Show steps
- Find a study group
- Meet with your study group regularly
- Review the material together
- Work on practice problems together
Calculus Workshop
Show steps
This activity will provide you with an opportunity to work with an expert calculus tutor and to receive personalized feedback on your work.
Browse courses on
Calculus
Show steps
Show steps
Show steps
- Attend a calculus workshop
- Work on practice problems with the tutor
- Receive feedback on your work
Calculus Poster
Show steps
Creating a poster can be a great way to summarize and review the key concepts of calculus.
Browse courses on
Calculus
Show steps
Show steps
Show steps
- Choose a topic for your poster
- Research your topic
- Create a visual representation of your topic
- Write a brief explanation of your topic
Contribute to Calculus Open Source Project
Show steps
Contributing to an open source calculus project can be a great way to deepen your understanding of the subject and to give back to the community.
Browse courses on
Calculus
Show steps
Show steps
Show steps
- Find an open source calculus project to contribute to
- Read the project documentation
- Make a contribution to the project
- Review and test your contribution
Career center
Learners who complete Calculus: Single Variable Part 3 - Integration will develop knowledge and skills
that may be useful to these careers:
Data Scientist
Data Scientist
A Data Scientist uses scientific methods to analyze large amounts of data in order to uncover meaningful insights. This course in Calculus provides you with the mathematical foundation needed to think critically about data, understand complex models, and make accurate predictions. The course's emphasis on conceptual understanding and applications will help you build a strong foundation for data science.
Financial Analyst
Financial Analyst
A Financial Analyst uses mathematical and statistical techniques to help companies make sound financial decisions. This Calculus course will provide you with the skills needed to understand financial models, analyze data, and make recommendations. The course's focus on problem-solving and critical thinking will help you succeed in this challenging field.
Quantitative Analyst
Quantitative Analyst
A Quantitative Analyst uses mathematical and statistical models to analyze financial data and make investment decisions. This Calculus course will provide you with the skills needed to understand financial models, analyze data, and make informed investment decisions. The course's emphasis on problem-solving and critical thinking will help you succeed in this competitive field.
Software Engineer
Software Engineer
A Software Engineer designs, develops, and maintains software systems. This Calculus course will provide you with the mathematical foundation needed to understand complex algorithms and data structures. The course's emphasis on problem-solving and critical thinking will help you succeed in this rapidly growing field.
Actuary
Actuary
An Actuary uses mathematical and statistical techniques to assess risk and uncertainty. This Calculus course will provide you with the skills needed to understand actuarial models, analyze data, and make informed decisions. The course's emphasis on problem-solving and critical thinking will help you succeed in this specialized field.
Operations Research Analyst
Operations Research Analyst
An Operations Research Analyst uses mathematical and statistical techniques to solve complex business problems. This Calculus course will provide you with the skills needed to understand optimization models, analyze data, and make informed decisions. The course's emphasis on problem-solving and critical thinking will help you succeed in this challenging field.
Statistician
Statistician
A Statistician uses mathematical and statistical techniques to collect, analyze, and interpret data. This Calculus course will provide you with the skills needed to understand statistical models, analyze data, and make informed decisions. The course's emphasis on problem-solving and critical thinking will help you succeed in this challenging field.
Economist
Economist
An Economist uses mathematical and statistical techniques to analyze economic data and make predictions about the economy. This Calculus course will provide you with the skills needed to understand economic models, analyze data, and make informed decisions. The course's emphasis on problem-solving and critical thinking will help you succeed in this challenging field.
Mathematician
Mathematician
A Mathematician uses mathematical and statistical techniques to solve complex problems. This Calculus course will provide you with the skills needed to understand mathematical models, analyze data, and make informed decisions. The course's emphasis on problem-solving and critical thinking will help you succeed in this challenging field.
Physicist
Physicist
A Physicist uses mathematical and statistical techniques to understand the laws of nature. This Calculus course will provide you with the skills needed to understand physical models, analyze data, and make predictions about the physical world. The course's emphasis on problem-solving and critical thinking will help you succeed in this challenging field.
Teacher
Teacher
A Teacher uses mathematical and statistical techniques to educate students about mathematics and its applications. This Calculus course will provide you with the skills needed to understand mathematical concepts, develop lesson plans, and teach students effectively. The course's emphasis on conceptual understanding and applications will help you succeed in this rewarding field.
Engineer
Engineer
An Engineer uses mathematical and statistical techniques to design and build structures and systems. This Calculus course will provide you with the skills needed to understand engineering models, analyze data, and make informed decisions. The course's emphasis on problem-solving and critical thinking will help you succeed in this challenging field.
Consultant
Consultant
A Consultant uses mathematical and statistical techniques to help businesses solve problems and make informed decisions. This Calculus course will provide you with the skills needed to understand business models, analyze data, and make recommendations. The course's emphasis on problem-solving and critical thinking will help you succeed in this challenging field.
Researcher
Researcher
A Researcher uses mathematical and statistical techniques to conduct research and develop new knowledge. This Calculus course will provide you with the skills needed to understand research methods, analyze data, and draw conclusions. The course's emphasis on critical thinking and problem-solving will help you succeed in this challenging field.
Analyst
Analyst
An Analyst uses mathematical and statistical techniques to analyze data and make recommendations. This Calculus course will provide you with the skills needed to understand analytical methods, analyze data, and make informed decisions. The course's emphasis on problem-solving and critical thinking will help you succeed in this challenging field.
For more career information including salaries, visit:
OpenCourser.com/course/erfkp2/calculus
Reading list
We've selected 35 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: Single Variable Part 3 - Integration.
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Rigorous and thorough introduction to calculus. It is written in a clear and concise style, and it covers all the topics in the course. It good choice for students who want to learn calculus in depth.
This classic textbook is known for its clear and concise explanations, as well as its wealth of solved examples and practice problems. It widely used resource for students and professionals in various fields that require a strong understanding of calculus.
Is comprehensive and thorough introduction to mathematical analysis. It covers all the foundations of calculus, and it is written in a clear and concise style. It good choice for students who want to learn calculus in a more theoretical way.
Save
Provides a clear and intuitive introduction to calculus, making it a great resource for students who are new to the subject or who want to review the basics. It also includes many helpful examples and exercises.
Rigorous and thorough introduction to real analysis. It covers the foundations of calculus, and it is written in a clear and concise style. It good choice for students who want to learn calculus in a more theoretical way.
Comprehensive and theoretical introduction to calculus. It covers all the topics in the course, and it is written in a clear and concise style. It good choice for students who want to learn calculus in a more theoretical way.
Save
Focuses specifically on integration and differential equations, which are key topics covered in the course. It offers a thorough treatment of these topics, with a strong emphasis on applications.
Save
Comprehensive and applied introduction to calculus. It covers all the topics in the course, and it is written in a clear and concise style. It good choice for students who want to learn calculus in a more applied way.
Classic textbook for calculus. It is well-written and covers all the topics in the course. It good choice for students who want a traditional approach to calculus.
Offers a conceptual approach to single-variable calculus, emphasizing the underlying ideas and applications of the subject. It is an excellent choice for learners who want to develop a strong foundation in the core concepts.
Combines the study of linear algebra and differential equations, providing a deep understanding of the relationship between these two branches of mathematics. It valuable resource for students who want to explore the connections between different mathematical concepts.
Comprehensive and applied introduction to calculus. It covers all the topics in the course, and it is written in a clear and concise style. It good choice for students who want to learn calculus in a more applied way.
Provides a comprehensive introduction to mathematical methods used in the physical sciences. It covers a wide range of topics, including vector analysis, partial differential equations, and applications in physics and engineering. It valuable resource for students who want to gain a deep understanding of the mathematical methods used in the physical sciences.
Save
Provides a thorough introduction to numerical methods for solving differential equations. It covers a variety of methods, including finite difference methods, finite element methods, and spectral methods. It valuable resource for students who want to learn how to apply numerical methods to real-world problems.
Save
History of calculus. It covers the development of calculus from its early beginnings to the present day. It good choice for students who want to learn more about the history of calculus.
Save
This comprehensive textbook that covers all of the major topics in calculus. It good choice for students who want a thorough understanding of the subject.
Save
Provides a comprehensive and accessible introduction to calculus. It covers a wide range of topics, from basic concepts to advanced applications. It great resource for students who want to explore the subject in depth.
Save
Is designed for students with little to no prior knowledge of calculus. It provides a gentle introduction to the subject, covering basic concepts and techniques. It great option for those who want to build a strong foundation before moving on to more advanced topics.
Provides an advanced treatment of ordinary differential equations. It great resource for students who want to delve deeper into the subject and gain a strong foundation for future research.
Provides a comprehensive introduction to complex variables and their applications. It covers a wide range of topics, including complex numbers, complex functions, and applications in physics and engineering. It valuable resource for students who want to explore the subject in depth.
Provides a comprehensive introduction to measure theory and integration. It covers a wide range of topics, including Lebesgue measure, integration theory, and applications. It valuable resource for students who want to gain a deep understanding of the foundations of integration.
Clear and concise introduction to calculus. It covers all the topics in the course, and it is written in a conversational style. It good choice for students who want to learn calculus in a more accessible way.
Provides a comprehensive introduction to the calculus of variations. It covers a wide range of topics, including the Euler-Lagrange equation, the Hamilton principle, and applications in physics and engineering. It valuable resource for students who want to gain a deep understanding of the calculus of variations.
Good choice for students who want a more concise and focused treatment of calculus. It covers the essential topics in a clear and engaging way.
Good choice for students who want a more theoretical treatment of differential equations. It covers a wide range of topics, from basic concepts to more advanced techniques.
Save
Covers a wide range of topics in applied mathematics, including calculus, linear algebra, and differential equations. It good choice for students who want a broad overview of the subject.
Good choice for students in engineering who need to learn the mathematics that is used in their field. It covers a wide range of topics, from basic concepts to more advanced techniques.
Covers a topic that is not covered in the course. However, it good choice for students who want to learn more about calculus on manifolds.
Covers a topic that is not covered in the course. However, it good choice for students who want to learn more about measure and integration.
Save
Covers a topic that is not covered in the course. However, it good choice for students who want to learn more about real analysis.
Covers a topic that is not covered in the course. However, it good choice for students who want to learn more about complex analysis.
Covers a topic that is not covered in the course. However, it good choice for students who want to learn more about partial differential equations.
Save
Covers a topic that is not covered in the course. However, it good choice for students who want to learn more about numerical analysis.
Covers a topic that is not covered in the course. However, it good choice for students who want to learn more about statistics and data analysis.
Covers a topic that is not covered in the course. However, it good choice for students who want to learn more about integration and ordinary differential equations.
For more information about how these books relate to this course, visit:
OpenCourser.com/course/erfkp2/calculus
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Effort
6-8 hours/week
Via
Coursera
Institution
University of Pennsylvania
Instructor
Robert Ghrist
Language
English
Transcripts
Español
Français
Pусский
Good to know
Develops essential Calculus skills, making it suitable for engineering, physical, and social science students
Introduces Taylor series and approximations at the outset, providing a robust mathematical foundation
Synthesizes discrete and continuous forms of Calculus, facilitating a comprehensive understanding
Emphasizes conceptual understanding over computational prowess, fostering a deep grasp of Calculus principles
Provides a clear, unified approach to Calculus, ensuring a coherent learning experience
The course is part of a five-part series, suggesting a comprehensive exploration of Calculus topics
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