The culmination point of the podcast well-defined & wonderful (for now, anyway) is the second part of the fundamental theorem. It combines the most important notions of the podcast so far: continuity, differentiation, and integration. We shall show that continuous functions on bounded and closed intervals always admit an anti-derivative. This anti-derivative is given as the integral of this function integrated with a variable right end point. The first and second part of the fundamental theorem lead to the substitution rule and the integration by parts formula. Finally we will also attempt to prove the second part of the fundamental theorem by providing the main tricks and glueing them together into one wonderful proof.

In this episode we are studying a first connection of differentiation and integration. More precisely, we will show that if a Riemann integrable function has an anti-derivative then the computation of the integral comes down to the evaluation of the anti-derivative. The proof provided uses a re-interpretation of the mean value theorem. A reorganisation of the terms involved in the statement of the mean value theorem leads to a relation of function evaluation and the integral of a step function with some height given by the derivative at some point of the function. A telescoping sum and a limit argument concludes the proof. 

This episode is focussing on a different sort of monotonicity compared to the notions we have used before. Here, we view the integral as a mapping assigning numbers to (Riemann integrable) functions. Monotonicity of the integral then means that non-negative functions are mapped to non-negative numbers. Or, in other words, if one function is smaller than another; their respective integrals can be compared the same way. In related contexts such mappings on functions are also called positive. As an application, we provide a fundamental inequality for the integral — a continuous variant of the triangle inequality: The modulus of the integral of a function is bounded above by integral of the modulus of the said function.