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