Lecture
This lecture provides several examples of applying the change of variables formula, starting with a region in the xy-plane and transforming it using different variable substitutions. The instructor demonstrates how to simplify integrals by changing to polar coordinates and discusses the challenges that arise when dealing with non-trivial domains. Through step-by-step calculations, the lecture covers various scenarios, including integrating over ellipses, circles, and more complex regions. The examples showcase the versatility of the change of variables technique in simplifying integration problems and highlight the importance of choosing suitable transformations for different types of domains.