Calculating Integrals in Multiple Dimensions

This lecture focuses on evaluating integrals in multiple dimensions, starting with the definition of integrals over regions in R2, R3, and beyond. The instructor explains the process of breaking down regions into rectangles, calculating approximate integrals, and moving towards the final integral. The lecture delves into the concept of building two-dimensional integrals from one-dimensional integrals, emphasizing the flexibility in the order of integration. Through a detailed proof, the instructor demonstrates how the integral over a rectangle can be expressed as a one-dimensional integral. The lecture further explores examples of calculating integrals over different regions, showcasing the interchangeability of integration orders in both two and three dimensions.