Calculating Integrals in Multiple Dimensions

This lecture focuses on evaluating integrals over nice regions in multiple dimensions, starting with rectangles in R2. The instructor explains how to calculate these integrals by developing a method similar to one-dimensional integration, using a step-by-step approach. The lecture covers the concept of continuous functions on rectangles, the definition of integrals, and the process of breaking down regions into smaller rectangles for calculation. The instructor demonstrates the calculation process through examples and emphasizes the interchangeability of integrating with respect to different variables. The lecture concludes with a discussion on the order of integration in three dimensions and provides examples to illustrate the concepts.