Chapter 15: Multiple Integrals

Multiple Integrals then covers double integrals over general regions, including type I and type II regions, and the change of variables to polar coordinates for circular symmetry problems. Applications include finding areas of regions, volumes under surfaces, average values of functions over regions, and mass calculations for planar laminae with variable density. The chapter advances to triple integrals for calculating volumes and mass in three dimensions, with evaluations in rectangular, cylindrical, and spherical coordinates. Change of variables is explored through the Jacobian determinant, enabling transformations to simplify complex regions. Students also apply multiple integrals to problems involving center of mass, moments of inertia, and probability density functions for multivariate distributions. Throughout, emphasis is placed on choosing coordinate systems that exploit symmetry and simplify computation. By the end, students can set up and evaluate multiple integrals in various coordinate systems, applying them to geometry, physics, engineering, and probability.