Calculus through Data & Modelling: Techniques of Integration from Coursera

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Syllabus

Module 1: Iterated Integrals

In this module, we extend the idea of a definite integral to double and even triple integrals of functions of two or three variables. These ideas are then used to compute areas, volumes, and masses of more general regions. Double integrals are also used to calculate probabilities when two random variables are involved. This extension of single variable calculus is the first step towards major tools that arise later in this specialization involving theorems of vector calculus.

Module 2: Double Integrals Over Plane Regions

For integrals of a function f(x), the region over which we integrate is always an interval of the real line. But for double integrals, we want to expand our abilities to integrate a multivariable function f(x,y) not only over rectangles, but also over more general regions in the plane. In this module, we develop the tools and techniques to do that.

Vector Functions

A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could be a scalar or a vector, but the output of this function is a vector. In this way, points are assigned to vectors. In this module, we will study these new types of functions and develop examples and applications of these new mathematical objects. They will play a key part in the development of vector calculus in future modules.

Integration with Data

Despite the broad algebraic tools we have learned to find antiderivatives and evaluate definite integrals using the Fundamental Theorem of Calculus, there are times when using antiderivatives is not possible. This could be because the function is too complicated in a way where no nice antiderivative exists, or that we are working with discrete data instead of a continuous function. In this module we introduce the notions and algorithms of numerical integration, which allow us to estimate the values of definite integrals. This is the basic problem we seek to solve: compute an approximate solution to a definite integral to a given degree of accuracy. There are many methods for approximating the integral to the desired precision, and we introduce a few here.

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

Develops the fundamentals of multivariable Calculus to aid with future vector Calculus

Provides a strong foundation for further coursework in vector Calculus

Suitable for learners with a background in single-variable Calculus

Applies multivariable Calculus to real-world problems, enhancing its relevance

Introduces numerical integration techniques to address challenges with finding antiderivatives

Taught by an experienced instructor with a PhD in the field

Reviews summary

Introductory calculus for undergrads

Learners largely praise Calculus through Data & Modelling: Techniques of Integration for being a well-made introduction to calculus for undergraduates. Quizzes and lectures from a wonderful lecturer are especially well received. The level of the course is considered easy for undergraduates. However, there are a few comments about the course lacking depth and having some topics out of order.

Students overwhelmingly praise this course.

"Best for undergraduates"

"I enjoyed completing Quizzes, however I'd be happy to see more practical tasks. Thank you! The lecturer is wonderful."

"Excellent course!"

Some students remark that topics were sometimes out of order.

"some topics were introduced a bit earlier than they should be."

Students describe this course as introductory.

"It was very straightforward and easy to understand but too superficial."

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: Techniques of Integration with these activities:

Review your notes and assignments from a previous calculus course

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Strengthen your foundation in calculus by reviewing key concepts from your previous coursework.

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Go through your notes and assignments from a previous calculus course.
Focus on the topics that will be covered in this course, such as limits, derivatives, and integrals.
Solve practice problems to test your understanding.

Read 'Vector Calculus' by Marsden

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Prepare yourself for the advanced concepts in vector calculus by reviewing some of the fundamental principles behind it.

View Vector Calculus on Amazon

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Read the first three chapters of the book.
Take notes on the key concepts, including vector fields, line integrals, and surface integrals.
Solve practice problems at the end of each chapter.

Read 'Calculus: Early Transcendentals' by Stewart

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Gain a comprehensive understanding of the foundational concepts in calculus by reviewing this well-regarded textbook.

View Algebra and Trigonometry on Amazon

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Read the relevant chapters on multivariable calculus.
Take notes on the key concepts, including partial derivatives, multiple integrals, and vector calculus.
Solve practice problems at the end of each chapter.

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Form a study group and discuss the concepts of vector-valued functions

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Enhance your understanding of vector-valued functions by collaborating with peers and exchanging perspectives.

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Find a group of classmates who are also enrolled in the course.
Meet regularly to discuss the course material, ask questions, and solve problems together.
Take turns presenting the material to each other.

Complete the Khan Academy module on Integrals in Two Dimensions

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Reinforce your understanding of iterated integrals and double integrals by completing this series of interactive tutorials.

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Watch the video lessons on each topic.
Complete the practice exercises.
Take the quizzes to test your understanding.

Solve 20 practice problems on double integrals involving regions defined by inequalities

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Build your proficiency in evaluating double integrals by completing a set of practice problems.

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Set up the integral in the correct form.
Evaluate the integral using the appropriate techniques.
Check your answers using a calculator or online tool.

Create a visual representation of a vector function

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Deepen your understanding of vector functions by creating a visual representation of one.

Browse courses on Vector Functions

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Choose a vector function to represent.
Use a graphing calculator or software to plot the vector field.
Analyze the resulting visualization to identify patterns and make observations.

Develop a computer program to calculate surface area and volume using triple integrals

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Apply your knowledge of triple integrals to a practical application by creating a tool that can compute surface area and volume.

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Design the algorithm for computing surface area and volume using triple integrals.
Implement the algorithm in a programming language of your choice.
Test the program on a variety of functions and compare the results to known values.

Career center

Learners who complete Calculus through Data & Modelling: Techniques of Integration will develop knowledge and skills that may be useful to these careers:

Operations Research Analyst

Operations Research Analysts develop, test, and implement mathematical and statistical techniques to help organizations operate more efficiently. Techniques from this course like approximating definite integrals with data will help Operations Research Analysts optimize and improve complex systems and processes.

See salaries and explore the career path for Operations Research Analyst

Data Scientist

Data Scientists are a cornerstone of the modern data economy and help organizations maximize the value of their data. The techniques utilized in this course, like double integrals on plane regions, can help build the foundation for a Data Scientist to compute areas, volumes, and masses of regions, which are foundational in this role.

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Financial Analyst

Financial Analysts utilize financial data to provide critical insights that shape business decisions. This course on Calculus through Data & Modelling can build the foundation for a Financial Analyst to apply integration techniques to evaluate complex financial models and make informed recommendations based on the data.

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Statistician

Statisticians play a critical role in extracting meaningful insights from data. This course in Calculus through Data & Modelling can help build the foundation for a Statistician, providing them with techniques like double and triple integrals to analyze and interpret complex datasets and draw evidence-based conclusions.

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Market Research Analyst

Market Research Analysts conduct research and analyze market trends and customer behavior to help businesses make informed decisions. This course on Calculus through Data & Modelling can provide a strong foundation for a Market Research Analyst, enabling them to apply data integration techniques to interpret complex market data and identify trends and opportunities.

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Actuary

Actuaries assess and manage risk and uncertainty in financial matters. This course in Calculus through Data & Modelling can provide a solid foundation for an Actuary. Techniques like integration and data modeling are crucial for analyzing risk, pricing insurance policies, and making sound financial decisions.

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Quantitative Analyst

Quantitative Analysts leverage mathematical and statistical models to analyze financial data and make investment decisions. Techniques learned in this course in Calculus through Data & Modelling, such as double integrals, provide foundational knowledge that Quantitative Analysts can use to evaluate complex financial instruments and make data-driven investment recommendations.

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Software Engineer

Software Engineers design, develop, and maintain software systems. This course in Calculus through Data & Modelling can be useful for Software Engineers who want to specialize in data-driven software solutions by providing a foundation in data integration and modeling techniques.

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Data Analyst

Data Analysts transform raw data into actionable insights to support business decision-making. This course on Calculus through Data & Modeling is valuable for Data Analysts, as it provides techniques like data integration and modeling that are essential for analyzing and interpreting large datasets.

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Economist

Economists study how people and societies make choices in the face of scarcity, and how these choices affect the allocation of resources. By providing techniques in data integration and modeling, this course on Calculus through Data & Modelling can support an Economist's understanding of economic data and market behaviors.

See salaries and explore the career path for Economist

Business Analyst

Business Analysts bridge the gap between business and technology, analyzing business processes and recommending solutions to improve efficiency. This course on Calculus through Data & Modelling can provide a foundation for Business Analysts, offering techniques like data integration and modeling that enable them to analyze data and identify business opportunities.

See salaries and explore the career path for Business Analyst

Risk Analyst

Risk Analysts identify, assess, and manage risks within an organization. This course on Calculus through Data & Modelling may be useful for Risk Analysts, as it provides a foundation in data integration and modeling techniques that can be applied to analyze, quantify, and mitigate risks.

See salaries and explore the career path for Risk Analyst

Quality Assurance Analyst

Quality Assurance Analysts ensure that products and services meet quality standards. This course on Calculus through Data & Modelling can provide a helpful foundation for Quality Assurance Analysts, as it offers techniques in data integration and modeling that can be used to analyze and improve quality control processes.

See salaries and explore the career path for Quality Assurance Analyst