Tobias Holck Colding continues MIT's 18.100B Real Analysis course by bounding the error in Taylor polynomial approximations. He works through how to estimate the remainder term that measures how closely a Taylor expansion tracks the original function, laying out the inequalities that make the bound rigorous. The second half of the lecture turns to building the Riemann integral from scratch, developing the formal machinery mathematicians use to define area under a curve for complicated functions. Colding frames integration as one of the central tools of mathematical analysis, tracing its origin to the practical problem of calculating areas of irregular figures before showing how the construction generalizes. The lecture is blackboard-based, with Colding working through definitions and proofs step by step, suited to students who have already covered earlier material in the course on limits and differentiation.