Sergei Yakovenko's blog: on Math and Teaching

Sunday, December 27, 2015

Lecture 9, Dec 22, 2015

Integral and antiderivative

  1. Area under the graph as a paradigm
  2. Definitions (upper and lower sums, integrability).
  3. Integrability of continuous functions.
  4. Newton-Leibniz formula: integral and antiderivative.
  5. Elementary rules of antiderivation (linearity, anti-Leibniz rule of “integration by parts”).
  6. Anti-chain rule, change of variables in the integral and its geometric meaning.
  7. Riemann–Stieltjes integral and change of variables in it.
  8. Integrability of discontinuous functions.

Not covered in the class: Lebesgue theorem and motivations for transition from Riemann to the Lebesgue integral.

The sketchy notes are available here.

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Wednesday, February 1, 2012

Last lecture: January 31, 2012

Integral: antiderivative and area

The last lecture (only partially exposed in the class) deals with the two seemingly unrelated problem: how to antidifferentiate functions (i.e., how to find a function when its derivative is known) and how to compute areas, in particular, under the graph of a given nonlinear function.

The answers turn out to be closely related by the famous Newton-Leibniz formula, which expresses the undergraph area through the antiderivative (primitive) of the function.

We discuss some tricks which allow to read the table of the derivatives from right to left (how to invert the Leibniz rule?) and find out that not all anterivatives can be “explicitly computed”. This “non-computability”, however, has its bright side: among “non-computable” antiderivatives we find functions which possess very special and useful properties, like the primitive of the power x^{-1}=\frac1x, which transforms multiplication into addition.

The lecture notes are available here.

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