Calculus 2, part 1 of 2: Integrals with applications
What you'll 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.
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 applications
Single variable calculus
S1. Introduction to the course
You 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 reverse
You 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 reverse
You 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 reverse
You 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 decomposition
You will learn: how to integrate rational functions using partial fraction decomposition.
S6. Trigonometric integrals
You 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 techniques
You 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 solving
You 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 properties
You 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 inspection
You 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 Calculus
You 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 curves
You 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 length
You will learn: compute the arc length of pieces of the graph of differentiable functions.
S14. Rotational volume
You will learn: compute various types of volumes with different methods.
S15. Surface area
You will learn: compute the area of surfaces obtained after rotation of pieces of the graph of differentiable functions.
S16. Improper integrals of the first kind
You will learn: evaluate integrals over infinite intervals.
S17. Improper integrals of the second kind
You 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 criteria
You will learn: using comparison criteria for determining convergence of improper integrals by comparing them to some well-known improper integrals.
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
Instructors
- 4.9 Instructor Rating
- 3,899 Reviews
- 31,088 Students
- 16 Courses
I am a multilingual mathematician with a passion for mathematics education. I always try to find the simplest possible explanations for mathematical concepts and theories, with illustrations whenever possible, and with geometrical motivations.
I worked as a senior lecturer in mathematics at Uppsala University (from August 2017 to August 2019) and at Mälardalen University (from August 2019 to May 2021) in Sweden, but I terminated my permanent employment to be able to create courses for Udemy full-time.
I am originally from Poland where I studied theoretical mathematics and got pedagogical qualifications at the Copernicus University in Toruń (1992-1997). Before that, I enjoyed a very rigorous mathematical education in a mathematical class in high school "Liceum IV" in Toruń, which gave me a very solid foundation for everything else I have learned and taught later.
In my courses I teach various branches of university mathematics that I have learned from absolutely excellent lectures of my dear professors from Toruń: Mirosław Uscki (b.1946), Zbigniew Bobiński (b.1940), Paweł Jarek (1933-2013), and Stanisław Balcerzyk (1932-2005).
My PhD thesis (2009) was at Uppsala University in Sweden, with the title: "Digital Lines, Sturmian Words, and Continued Fractions".
In 2018 I received four pedagogical prizes from students at the Faculty of Science and Technology of Uppsala University: on May 13th from the students at the Master Program in Engineering Physics; on May 25th from the students at the Master Program in Electrical Engineering; on December 20th from the students at the Master Program in Chemical Engineering; on January 10th 2019 from UTN (Uppsala Union of Engineering and Science Students at Uppsala University).
I speak Polish, Swedish, English, Dutch, and some Russian; learning Ukrainian.
- 4.9 Instructor Rating
- 3,899 Reviews
- 31,088 Students
- 16 Courses
I have a background in medicine and software development. I've done enough mathematics to at least follow along in Hania's courses and I'm learning a lot as I edit the material. I have also written a book about medical software design as it pertains to the medical record ("Rethinking the electronic healthcare record"). For Hania's math courses, it's my job to set up the environment and produce the final output that goes into these courses.