12.10.2024, 00:47
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Calculus 2, part 1 of 2: Integrals with applications
Published 10/2024
Created by Hania Uscka-Wehlou,Martin Wehlou
MP4 | Video: h264, 1280x720 | Audio: AAC, 44.1 KHz, 2 Ch
Genre: eLearning | Language: English | Duration: 262 Lectures ( 56h 46m ) | Size: 46.4 GB
Integral calculus in one variable: theory and applications for computing area between curves, curve length, and volumes
[b]What you'll learn:[/b]
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.
Requirements:
Precalculus (Basic notions, Polynomials and rational functions, Trigonometry, Exponentials and logarithms)
Calculus 1: Limits and continuity (or equivalent)
Calculus 1: Derivatives with applications (or equivalent)
You are always welcome with your questions. If something in the lectures is unclear, please, ask. It is best to use QA, so that all the other students can see my additional explanations about the unclear topics. Remember: you are never alone with your doubts, and it is to everybody's advantage if you ask your questions on the forum.
Description:
Calculus 2, part 1 of 2: Integrals with applicationsSingle variable calculusS1. Introduction to the courseYou will learn: about the content of this course and about importance of Integral Calculus. The purpose of this section is not to teach you all the details (this comes later in the course) but to show you the big picture.S2. Basic formulas for differentiation in reverseYou will learn: the concept of antiderivative (primitive function, indefinite integral); formulas for the derivatives of basic elementary functions in reverse.S3. Integration by parts: Product Rule in reverseYou will learn: understand and apply the technique of integration called "integration by parts"; some very typical and intuitively clear examples (sine or cosine times a polynomial, the exponential function times a polynomial), less obvious examples (sine or cosine times the exponential function), mind-blowing examples (arctangent and logarithm), and other examples.S4. Change of variables: Chain Rule in reverseYou will learn: how to perform variable substitution in integrals and how to recognise that one should do just this.S5. Integrating rational functions: partial fraction decompositionYou will learn: how to integrate rational functions using partial fraction decomposition.S6. Trigonometric integralsYou will learn: how to compute integrals containing trigonometric functions with various methods, like for example using trigonometric identities, using the universal substitution (tangent of a half angle) or other substitutions that reduce our original problem to the computing of an integral of a rational function.S7. Direct and inverse substitution, and more integration techniquesYou will learn: Euler substitutions; the difference between direct and inverse substitution; triangle substitutions (trigonometric substitutions); some alternative methods (by undetermined coefficients) in cases where we earlier used integration by parts or variable substitution.S8. Problem solvingYou will learn: you will get an opportunity to practice the integration techniques you have learnt until now; you will also get a very brief introduction to initial value problems (topic that will be continued in a future ODE course, Ordinary Differential Equations).S9. Riemann integrals: definition and propertiesYou will learn: how to define Riemann integrals (definite integrals) and how they relate to the concept of area; partitions, Riemann (lower and upper) sums; integrable functions; properties of Riemann integrals; a proof of uniform continuity of continuous functions on a closed bounded interval; a proof of integrability of continuous functions (and of functions with a finite number of discontinuity points); monotonic functions; a famous example of a function that is not integrable; a formulation, proof and illustration of The Mean Value Theorem for integrals; mean value of a function over an interval.S10. Integration by inspectionYou will learn: how to determine the value of the integrals of some functions that describe known geometrical objects (discs, rectangles, triangles); properties of integrals of even and odd functions over intervals that are symmetric about the origin; integrals of periodic functions.S11. Fundamental Theorem of CalculusYou will learn: formulation, proof and interpretation of The Fundamental Theorem of Calculus; how to use the theorem for: 1. evaluating Riemann integrals, 2. computing limits of sequences that can be interpreted as Riemann sums of some integrable functions, 3. computing derivatives of functions defined with help of integrals; some words about applications of The Fundamental Theorem of Calculus in Calculus 3 (Multivariable Calculus).S12. Area between curvesYou will learn: compute the area between two curves (graphs of continuous functions), in particular between graphs of continuous functions and the x-axis.S13. Arc lengthYou will learn: compute the arc length of pieces of the graph of differentiable functions.S14. Rotational volumeYou will learn: compute various types of volumes with different methods.S15. Surface areaYou will learn: compute the area of surfaces obtained after rotation of pieces of the graph of differentiable functions.S16. Improper integrals of the first kindYou will learn: evaluate integrals over infinite intervals.S17. Improper integrals of the second kindYou will learn: evaluate integrals over intervals that are not closed, where the integrand can be unbounded at (one or both of) the endpoints.S18. Comparison criteriaYou will learn: using comparison criteria for determining convergence of improper integrals by comparing them to some well-known improper integrals.S19. ExtrasYou will learn: about all the courses we offer. You will also get a glimpse into our plans for future courses, with approximate (very hypothetical!) release dates.Make sure that you check with your professor what parts of the course you will need for your final exam. Such things vary from country to country, from university to university, and they can even vary from year to year at the same university.A detailed description of the content of the course, with all the 261 videos and their titles, and with the texts of all the 419 problems solved during this course, is presented in the resource file 001 List_of_all_Videos_and_Problems_Calculus_2_p1.pdf under video 1 ("Introduction to the course"). This content is also presented in video 1.
Who this course is for:
University and college students wanting to learn Single Variable Calculus (or Real Analysis)
High school students curious about university mathematics; the course is intended for purchase by adults for these students
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