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Joseph W. Cutrone, PhD
This course continues your study of calculus by focusing on the applications of integration. The applications in this section have many common features. First, each is an example of a quantity that is computed by evaluating a definite integral. Second, the formula for that application is derived from Riemann sums.
Read more
This course continues your study of calculus by focusing on the applications of integration. The applications in this section have many common features. First, each is an example of a quantity that is computed by evaluating a definite integral. Second, the formula for that application is derived from Riemann sums.
Read more
This course continues your study of calculus by focusing on the applications of integration. The applications in this section have many common features. First, each is an example of a quantity that is computed by evaluating a definite integral. Second, the formula for that application is derived from Riemann sums.
Rather than measure rates of change as we did with differential calculus, the definite integral allows us to measure 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.
We will expand the notion of the average value of a data set to allow for infinite values, develop the formula for arclength and curvature, and derive formulas for velocity, acceleration, and areas between curves. Through examples and projects, we will apply the tools of this course to analyze and model real world data.
Register for this course and see more details by visiting:
OpenCourser.com/course/yi1tik/calculus
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What's inside
Syllabus
Module 1: Average Value of a Function
In this module, we generalize the notion of the average value of a (finite) set of points. Did you ever wonder how we compute the average temperature during the day if infinitely many temperature readings are possible? Or how the average rainfall is calculated? The notions in this module will allow us to expand the idea of an average value to compute averages with (infinite) values over a continuous interval.
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Syllabus
Module 1: Average Value of a Function
In this module, we generalize the notion of the average value of a (finite) set of points. Did you ever wonder how we compute the average temperature during the day if infinitely many temperature readings are possible? Or how the average rainfall is calculated? The notions in this module will allow us to expand the idea of an average value to compute averages with (infinite) values over a continuous interval.
Read more
Traffic lights
Traffic lights
Read about what's good
what should give you pause and
possible dealbreakers
Covers applications of integration, a core calculus concept
Expands the notion of the average value of a function to allow for infinite values
Develops formulas for arc length, curvature, velocity, acceleration, and areas between curves
Applies calculus tools to analyze and model real-world data
Provides examples and projects for practical application
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Reviews summary
Applied integration for data & modeling
According to students, this course is highly effective in bridging theoretical calculus with practical applications, particularly for those in quantitative fields. Learners consistently praise its focus on real-world data and modeling across diverse applications like economics, engineering, and environmental science, making concepts highly relevant. Many find the instructor's explanations clear and engaging, especially when deriving formulas from Riemann sums, which deepens understanding. While the course offers challenging yet rewarding projects, some learners with insufficient prerequisite knowledge found the pace inconsistent or explanations occasionally too abstract, necessitating additional study.
Projects reinforce learning and are seen as highly rewarding.
"The projects were challenging yet incredibly rewarding."
"Through examples and projects, we will apply the tools of this course to analyze and model real world data."
"The hands-on projects reinforced the learning beautifully."
Instructor clarity and focus on derivations deepen understanding.
"The instructor's passion for the subject is evident, making even difficult topics engaging."
"The emphasis on deriving formulas from Riemann sums really makes you understand the 'why' behind the integral applications."
"The instructor is very knowledgeable and explains concepts clearly."
Course excels in connecting theory to real-world data and scenarios.
"This course was truly insightful, bridging theoretical calculus with practical applications in data modeling."
"Absolutely fantastic! As an engineer, the practical applications were invaluable."
"The examples relating to population and environmental data were very relevant."
Some found the pace uneven or felt need for more in-depth examples.
"My only minor gripe is that sometimes the pace felt a bit slow in introductory parts and then very fast in the advanced ones."
"I struggled with some of the derivations, and felt that the instructor assumed too much prior knowledge for certain topics."
"I wish there were more step-by-step examples."
Requires a solid calculus foundation; not ideal for beginners.
"I found this course quite difficult to follow. ... Not for beginners."
"The prerequisite knowledge is definitely a must; without it, you'll struggle."
"I'd recommend it to those with a strong calculus foundation."
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: Integration Applications with these
activities:
Review integration by parts
Show steps
Review integration by parts to brush up on techniques that are essential for succeeding in this course.
Browse courses on
Integration by Parts
Show steps
-
Revisit the formula for integration by parts
-
Work through practice problems using integration by parts.
-
Complete a practice quiz on integration by parts.
Organize and review course materials
Show steps
Keep your course materials well-organized and review them regularly to enhance retention and understanding.
Show steps
-
Create a system for organizing notes, assignments, quizzes, and exams.
-
Review your organized materials on a regular basis.
Tutorials on vector calculus
Show steps
Explore online tutorials to reinforce your understanding of vector calculus concepts covered in this course.
Browse courses on
Vector Calculus
Show steps
-
Identify online tutorials that cover relevant vector calculus topics.
-
Work through the tutorials at your own pace.
-
Complete practice exercises or quizzes provided by the tutorials.
Four other activities
Expand to see all activities and additional details
Show all seven activities
Form a study group with classmates
Show steps
Collaborate with peers to discuss course concepts, work on problems together, and reinforce your understanding.
Show steps
-
Identify classmates who are interested in forming a study group.
-
Establish regular meeting times and a study schedule.
-
Take turns leading discussions and presenting solutions.
Practice problems on arc length and curvature
Show steps
Solve numerous practice problems to master the concepts of arc length and curvature covered in this course.
Show steps
-
Gather practice problems from textbooks, online resources, or the course instructor.
-
Allocate dedicated time for solving the practice problems.
-
Review your solutions and identify areas for improvement.
Create a visualization tool for velocity and acceleration
Show steps
Develop a visual representation of velocity and acceleration to enhance your understanding of these concepts.
Browse courses on
Velocity
Show steps
-
Choose a programming language or software for creating visualizations.
-
Design the visualization tool, including the input and output formats.
-
Implement the visualization tool and test its functionality.
-
Share your visualization tool with others for feedback and improvement.
Contribute to an open-source project related to calculus
Show steps
Engage with the open-source community by contributing to a project that aligns with the concepts covered in this course.
Browse courses on
Calculus
Show steps
-
Identify an open-source project related to calculus.
-
Review the project's documentation and contribute code or documentation.
-
Collaborate with other contributors and seek feedback on your contributions.
Review integration by parts
Show steps
Review integration by parts to brush up on techniques that are essential for succeeding in this course.
Browse courses on
Integration by Parts
Show steps
Show steps
Show steps
- Revisit the formula for integration by parts
- Work through practice problems using integration by parts.
- Complete a practice quiz on integration by parts.
Organize and review course materials
Show steps
Keep your course materials well-organized and review them regularly to enhance retention and understanding.
Show steps
Show steps
Show steps
- Create a system for organizing notes, assignments, quizzes, and exams.
- Review your organized materials on a regular basis.
Tutorials on vector calculus
Show steps
Explore online tutorials to reinforce your understanding of vector calculus concepts covered in this course.
Browse courses on
Vector Calculus
Show steps
Show steps
Show steps
- Identify online tutorials that cover relevant vector calculus topics.
- Work through the tutorials at your own pace.
- Complete practice exercises or quizzes provided by the tutorials.
Four other activities
Expand to see all activities and additional details
Show all seven activities
Form a study group with classmates
Show steps
Collaborate with peers to discuss course concepts, work on problems together, and reinforce your understanding.
Show steps
Show steps
Show steps
- Identify classmates who are interested in forming a study group.
- Establish regular meeting times and a study schedule.
- Take turns leading discussions and presenting solutions.
Practice problems on arc length and curvature
Show steps
Solve numerous practice problems to master the concepts of arc length and curvature covered in this course.
Show steps
Show steps
Show steps
- Gather practice problems from textbooks, online resources, or the course instructor.
- Allocate dedicated time for solving the practice problems.
- Review your solutions and identify areas for improvement.
Create a visualization tool for velocity and acceleration
Show steps
Develop a visual representation of velocity and acceleration to enhance your understanding of these concepts.
Browse courses on
Velocity
Show steps
Show steps
Show steps
- Choose a programming language or software for creating visualizations.
- Design the visualization tool, including the input and output formats.
- Implement the visualization tool and test its functionality.
- Share your visualization tool with others for feedback and improvement.
Contribute to an open-source project related to calculus
Show steps
Engage with the open-source community by contributing to a project that aligns with the concepts covered in this course.
Browse courses on
Calculus
Show steps
Show steps
Show steps
- Identify an open-source project related to calculus.
- Review the project's documentation and contribute code or documentation.
- Collaborate with other contributors and seek feedback on your contributions.
Career center
Learners who complete Calculus through Data & Modelling: Integration Applications will develop knowledge and skills
that may be useful to these careers:
Data Scientist
Data Scientist
Data Scientists combine scientific methods, programming skills, and statistical techniques to extract insights from data. By learning about integration applications, you can develop the skills needed to build predictive models, analyze complex datasets, and solve real-world problems using data-driven approaches.
Data Analyst
Data Analyst
A Data Analyst specializes in analyzing data to uncover trends and patterns that can help businesses make better decisions. By learning about integration applications, you can develop the skills needed to collect, analyze, and interpret data to draw meaningful conclusions and support data-driven decision-making in this role.
Statistician
Statistician
Statisticians collect, analyze, interpret, and present data. By studying integration applications, you can gain a strong foundation in statistical methods and techniques, which are essential for designing studies, analyzing data, and drawing meaningful conclusions from statistical evidence.
Operations Research Analyst
Operations Research Analyst
Operations Research Analysts apply mathematical and analytical techniques to solve complex problems and improve efficiency in business operations. By studying integration applications, you can build a foundation for developing optimization models and making data-driven decisions that can streamline processes and enhance performance.
Risk Analyst
Risk Analyst
Risk Analysts assess and manage risks for businesses and organizations. By studying integration applications, you can develop the analytical skills and understanding of probability and uncertainty needed to evaluate risks, develop mitigation strategies, and make informed decisions in the face of uncertainty.
Actuarial Analyst
Actuarial Analyst
Actuarial Analysts use mathematical and statistical techniques to assess financial risks for insurance companies and other financial institutions. The concepts of integration applications covered in this course can provide you with a strong foundation for understanding actuarial models and making informed decisions in managing risks and uncertainties.
Financial Analyst
Financial Analyst
Financial Analysts use mathematical and statistical techniques to analyze financial data and make recommendations on investments and other financial decisions. The concepts of integration applications covered in this course can provide you with a strong foundation for understanding financial models and making sound investment decisions.
Financial Engineer
Financial Engineer
Financial Engineers develop and implement mathematical models for financial applications. The knowledge gained in integration applications can provide you with a solid foundation in financial mathematics and modeling, which are essential for developing innovative financial products and strategies.
Research Analyst
Research Analyst
Research Analysts conduct research and provide insights on a wide range of topics, including economics, finance, politics, and healthcare. By studying integration applications, you can develop the analytical skills and ability to interpret complex data, which are critical for producing high-quality research reports and providing valuable insights.
Economist
Economist
Economists study how societies allocate resources and make decisions. The skills you gain in integration applications can help you develop a deep understanding of economic models and theories, which are essential for analyzing economic data, forecasting economic trends, and developing economic policies.
Market Research Analyst
Market Research Analyst
Market Research Analysts conduct research to understand consumer behavior, market trends, and industry dynamics. The skills you gain in integration applications can help you analyze market data, identify market opportunities, and develop strategies to target specific customer segments.
Biostatistician
Biostatistician
Biostatisticians apply statistical principles to analyze biological and medical data. By studying integration applications, you can develop the skills needed to design and conduct statistical studies, interpret complex data, and contribute to the advancement of medical research and public health initiatives.
Quantitative Analyst
Quantitative Analyst
Quantitative Analysts develop and use mathematical and statistical models to analyze financial markets and make investment decisions. This course on integration applications can provide you with the analytical tools and understanding of mathematical concepts needed to succeed in quantitative finance.
Actuary
Actuary
Actuaries assess and manage financial risks for insurance companies and other financial institutions. The knowledge gained in integration applications can provide you with a strong foundation for understanding actuarial models and making informed decisions in managing risks and uncertainties.
Software Developer
Software Developer
Software Developers design, develop, and maintain software systems. The concepts of integration applications covered in this course can help you build a solid foundation in mathematical algorithms and data structures, which are essential for developing efficient and reliable software solutions.
For more career information including salaries, visit:
OpenCourser.com/course/yi1tik/calculus
Reading list
We've selected 13 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: Integration Applications.
This classic calculus textbook provides a comprehensive overview of the material covered in the course, including applications of integration. It valuable resource for students who want to review the basics or learn more about the subject.
Save
Provides a thorough treatment of calculus, including topics such as the Riemann sum and the definite integral. It good choice for students who want a deeper understanding of the mathematical concepts behind the course material.
Provides a comprehensive overview of calculus, with a focus on applications. It good choice for students who want to learn how to use calculus to solve problems in areas such as physics, engineering, and economics.
This classic textbook provides a rigorous treatment of calculus. It good choice for students who want a deep understanding of the mathematical concepts behind the course material.
Save
Provides a concise and rigorous introduction to calculus. It good choice for students who want to learn the basics of calculus quickly and efficiently.
Save
Provides a unique and intuitive approach to calculus. It good choice for students who want to learn calculus in a way that emphasizes its physical and geometric applications.
Provides a historical perspective on the development of calculus. It good choice for students who want to learn about the origins of calculus and how it has been used to solve problems throughout history.
Provides a graphical approach to calculus. It good choice for students who want to learn calculus in a way that emphasizes its visual and intuitive aspects.
Provides a computational approach to calculus. It good choice for students who want to learn how to use calculus to solve problems using computers.
Save
Provides a modeling approach to calculus. It good choice for students who want to learn how to use calculus to create and analyze mathematical models of real-world problems.
Save
Provides a problem-solving approach to calculus. It good choice for students who want to learn how to use calculus to solve a wide variety of problems.
Save
Provides a new approach to calculus. It good choice for students who want to learn calculus in a way that emphasizes its conceptual and theoretical foundations.
Provides a gentle introduction to calculus. It good choice for students who want to learn calculus in a relaxed and non-threatening environment.
For more information about how these books relate to this course, visit:
OpenCourser.com/course/yi1tik/calculus
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Johns Hopkins University
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Joseph W. Cutrone, PhD
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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
- Calculus through Data & Modelling: Integration Applications (viewing)
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