Tobias Holck Colding continues MIT's 18.100B Real Analysis course with a lecture on the fundamental theorem of calculus, the result linking differentiation and integration. He presents two versions of the theorem, working through the rigorous conditions under which each holds, then moves into improper integrals and integrals of unbounded functions. The eighty minute session builds on earlier lectures in the course's treatment of Riemann integration, proving the statements carefully rather than just stating them, in keeping with the course's real analysis approach. This is lecture nineteen of the Spring 2025 offering, delivered as a chalkboard proof-based session aimed at students who already have the groundwork in limits, continuity, and differentiation from earlier in the term.