Calculus 2, part 1 of 2: Integrals with applications

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.

S19. Extras

You 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.