Calculus through Data & Modelling: Series and Integration from Coursera

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.

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.

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.

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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

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

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Establish a system for organizing your notes, assignments, and study materials to enhance your ability to review and retain information.

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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

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Ensure you have a strong foundation in algebra and trigonometry, which are essential prerequisites for calculus.

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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

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Strengthen your understanding of the fundamentals of integration by reviewing a comprehensive textbook.

View Algebra and Trigonometry on Amazon

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Read through the chapters covering integration.
Take notes and highlight important concepts.
Work through the practice problems at the end of each chapter.

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Explore Online Tutorials on Integration

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Supplement your learning by seeking out online tutorials that provide additional explanations and examples of integration techniques.

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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

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Collaborate with peers to reinforce your understanding of integration and enhance your problem-solving skills.

Browse courses on Integration

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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

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Deepen your understanding of integration techniques by solving a variety of practice problems.

Browse courses on Indefinite Integrals

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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

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Enhance your understanding and ability to explain integration by creating a visual guide that illustrates key concepts.

Browse courses on Fundamental Theorem of Calculus

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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

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Enhance your understanding and application of integration by working on a project that involves modeling a real-world problem.

Browse courses on Calculus

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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 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.

See salaries and explore the career path for Data 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.

See salaries and explore the career path for Financial 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.

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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.

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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.

See salaries and explore the career path for Statistician

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.

See salaries and explore the career path for Quantitative Analyst

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.

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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.

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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.

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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.

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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.

See salaries and explore the career path for Economist

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.

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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.

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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.

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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.

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