Calculus 2, part 1 of 2: Integrals with applications
Integral calculus in one variable: theory and applications for computing area between curves, curve length, and volumes
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Oct 2024
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What you will learn
How to solve problems concerning integrals of real-valued functions of 1 variable (illustrated with 419 solved problems) and why these methods work.
The concept of antiderivative / primitive function / indefinite integral of a function, and computing such integrals in a process reverse to differentiation.
Integration by parts as the Product Rule in reverse with many examples of its applications.
Integration by substitution as the Chain Rule in reverse with many examples of its applications.
Integration of rational functions with help of partial fraction decomposition.
Various types of trigonometric integrals and how to handle them.
Direct and inverse substitutions; various types of trigonometric substitutions.
The tangent half-angle substitution (universal trigonometric substitution).
Euler's substitutions.
Triangle substitutions.
Riemann integral (definite integral): its definition and geometrical interpretation in terms of area.
An example of a function that is not Riemann integrable (the characteristic function of the set Q, restricted to [0,1]).
Oscillatory sums; Cauchy criterion of (Riemann) integrability.
Sequential characterisation of (Riemann) integrability.
Proof of uniform continuity of continuous functions on a closed bounded interval.
Integrability of continuous functions on closed intervals.
Integration by inspection: Riemann integrals of odd (or: even) functions over compact and symmetric-to-zero intervals.
Integration by inspection: evaluating some definite integrals with help of areas known from geometry.
Fundamental Theorem of Calculus (FTC) in two parts, with a proof.
Applications of Fundamental Theorem of Calculus in Calc 2 and Calc3.
Application of FTC for computing derivatives of functions defined with help of Riemann integrals with variable (one or both) limits of integration.
Application of FTC for computing limits of sequences that can be interpreted as Riemann sums for some integrable functions.
The Mean-Value Theorem for integrals with proof and with a geometrical interpretation; the concept of a mean value of a function on an interval.
Applications of Riemann integrals: (signed) area between graphs of functions and the x-axis, area between curves defined by two continuous functions.
Applications of Riemann integrals: rotational volume.
Applications of Riemann integrals: rotational area.
Applications of Riemann integrals: curve length.
Improper integrals of the first kind (integration over an unbounded interval).
Improper integrals of the second kind (integration of unbounded functions).
Comparison criteria for determining whether an improper integral is convergent or not.
Why take this course?
🚀 Calculus 2, Part 1 of 2: Integrals with Applications 📚
Course Introduction 🎓
Dive into the world of Integral Calculus, where we'll explore the theory and practical applications that allow us to compute areas between curves, curve lengths, and volumetric measurements. This course is a pivotal step in your journey through advanced calculus.
🔹 What You'll Learn
- The scope of this course and the significance of Integral Calculus.
- The basics of differentiation in reverse to understand integrals as a process of reversing the differentiation process.
Integration by Parts: Product Rule in Reverse 🍏⫫📈
- ⭐️ Master Integration by Parts: Learn this fundamental technique and see it applied to various functions, from basic to mind-bending examples.
Change of Variables: Chain Rule in Reverse 🌍
- ⭐️ Solve Complex Integrals: Discover how variable substitution can simplify multistep problems within integrals.
Integrating Rational Functions: Partial Fraction Decomposition 🎢
- ⭐️ Break Down Rational Functions: Learn the art of breaking down rational functions into simpler parts for easier integration.
Trigonometric Integrals ⚫️
- ⭐️ Conquer Trigonometric Functions: Learn different methods to compute integrals with trigonometric functions, including some exotic ones!
The Fundamental Theorem of Calculus & Beyond 🧭
- ⭐️ Understand The Fundamental Theorem: Explore its applications in evaluating areas, computing limits, and finding derivatives.
Area, Arc Length, Rotational Volume & More 📏✨
- ⭐️ Calculate Geometric Properties: Compute the area between curves, arc length of functions, rotational volume, and even more.
Improper Integrals: Exploring the Infinite ⚡
- ⭐️ Tackle Improper Integrals: Learn about improper integrals over infinite intervals and at endpoint discontinuities.
Comparison Criteria for Convergence 🔍
- ⭐️ Evaluate Convergence: Use comparison criteria to determine the convergence of improper integrals by comparing them to well-known cases.
Extras & What's Next? 👀
- ⭐️ Explore More: Get a sneak peek into additional courses we offer and catch a glimpse of our future course plans.
Course Resources 📚
- A comprehensive list of all videos and problems for this course is available in the resource file "001 List_of_all_Videos_and_Problems_Calculus_2_p1.pdf", located under video 1, "Introduction to the Course."
Make sure to align your studies with your professor's expectations for the final exam. The content of this course can vary by region and institution, and even from year to year.
Embark on a calculus odyssey that will transform the way you see geometry, analysis, and problem-solving. With Integral Calculus as your compass, navigate through the intricate landscape of advanced mathematics! 🌟
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udemy ID
03/03/2024
course created date
09/10/2024
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