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Calculus through Data & Modelling

Integration Applications

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.

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.

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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.

Module 2: Arc Length and Curvature

What do we mean by the arc length of a curve? We might think of fitting a piece of string to the curve and then measuring the string against a ruler. But this is difficult to do when working with a complicated curve. In this module we develop the precise notion of the length and curvature of an arc of a curve in both the xy plane and in space.

Module 4: Velocity and Acceleration

In this module, we show how the ideas of tangent and normal vectors can be used in physics to study the motion of an object, including its velocity and acceleration, but now we focus on curves in three dimensional space. The techniques developed here then allow us to study the rates of change for more advanced functions.

Module 4: Areas Between Curves

Finding the area between two curves is not just an interesting application of definite integrals from a geometric view, but when working with the appropriate functions, has applications in economics, business, and even medicine.

Good to know

Know what's good

, what to watch for

, 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

Challenging calculus course

Learners say this Calculus course is challenging, with overly simplistic lectures and difficult exams. However, the course is well received because it is well-balanced in the first half of the specialization, and concepts like curvature are clearly explained with engaging assignments.

This course has engaging assignments.

The first half of the course is well balanced.

"The first half of this specialization is ok and well balanced."

Curvature is clearly explained.

"質点运动的概念李俊峰的教材可以整理清楚"

This course is challenging.

"Sadly, in this course the material was presented in such an oversimplified way and stripped of details that it was very difficult to develop an understanding of the concepts being presented."

"The first half of this specialization is ok and well balanced. But in my mind, this second half, specially since calculus, the complexity and practice of classes is umbalance with the complexity of the test."

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

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Review integration by parts to brush up on techniques that are essential for succeeding in this course.

Browse courses on Integration by Parts

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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

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Keep your course materials well-organized and review them regularly to enhance retention and understanding.

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Create a system for organizing notes, assignments, quizzes, and exams.
Review your organized materials on a regular basis.

Tutorials on vector calculus

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Explore online tutorials to reinforce your understanding of vector calculus concepts covered in this course.

Browse courses on Vector Calculus

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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

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Show all seven activities

Form a study group with classmates

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Collaborate with peers to discuss course concepts, work on problems together, and reinforce your understanding.

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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

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Solve numerous practice problems to master the concepts of arc length and curvature covered in this course.

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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

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Develop a visual representation of velocity and acceleration to enhance your understanding of these concepts.

Browse courses on Velocity

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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

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Engage with the open-source community by contributing to a project that aligns with the concepts covered in this course.

Browse courses on Calculus

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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 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.

See salaries and explore the career path for Data Scientist

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.

See salaries and explore the career path for Data Analyst

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.

See salaries and explore the career path for Statistician

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.

See salaries and explore the career path for Risk 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.

See salaries and explore the career path for Operations Research 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.

See salaries and explore the career path for Actuarial 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.

See salaries and explore the career path for Financial Analyst

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.

See salaries and explore the career path for Financial Engineer

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.

See salaries and explore the career path for 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.

See salaries and explore the career path for Market Research Analyst

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.

See salaries and explore the career path for Economist

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.

See salaries and explore the career path for Biostatistician

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.

See salaries and explore the career path for Quantitative Analyst

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.

See salaries and explore the career path for Actuary

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.

See salaries and explore the career path for Software Developer

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.

Algebra and Trigonometry

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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.

Calculus through Data & Modelling

Integration Applications

Three deals to help you save

What's inside

Syllabus

Good to know

Save this course

Reviews summary

Challenging calculus course

Activities

Career center

Reading list

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