Change of Variables in 3D: Cylindrical Coordinates

This lecture covers classic variable changes in 3D, focusing on the domain in R3D defined by X² + Y² < 1, 0 < Z < 1. It introduces cylindrical coordinates as a way to describe cylinders in R3, with U ranging from 0 to 1, V from 0 to 2π, and WZ from 0 to 1-U. The Jacobian is calculated for the transformation, leading to the integration of functions like X² + Y² in the new coordinate system. Through examples, the instructor demonstrates how to integrate functions using the Jacobian and cylindrical coordinates, showcasing the application of these concepts in calculating volumes of shapes like cones.