Integration allows us to find volumes of three-dimensional solids by breaking them into infinitesimally thin pieces and adding them up. When we rotate a two-dimensional region around an axis, we create a solid of revolution. The key insight is that we can slice this solid into thin disks or shells, calculate each piece's volume, and integrate to find the total volume.
The disk method is used when we rotate a region around an axis and slice the resulting solid perpendicular to that axis. Each slice forms a circular disk. The radius of each disk equals the function value at that point. The volume of a thin disk is pi times radius squared times thickness. We integrate this expression over the interval to find the total volume.
The shell method is an alternative approach for finding volumes of solids of revolution. Instead of slicing perpendicular to the axis, we slice parallel to it, creating cylindrical shells. Each shell has radius x, height f of x, and thickness dx. The volume of a shell is two pi times radius times height times thickness. This method is often easier when rotating around the y-axis.
The cross-sectional area method is used when we have a solid with a known base and cross-sections of known shape. For example, if we have a circular base and square cross-sections perpendicular to the x-axis, we find the area function A of x for each cross-section. The volume is the integral of this area function over the interval. This method works for any shape of cross-section: triangles, semicircles, or any geometric figure.
In summary, integration provides three powerful methods for finding volumes. The disk method slices perpendicular to the axis of rotation, the shell method slices parallel to it, and the cross-sectional area method works with any known cross-section shape. These techniques have wide applications in engineering for designing containers and structures, in physics for calculating moments of inertia, and in architecture for planning complex geometries. Mastering these methods gives you the tools to solve virtually any volume problem.