Chapter 6: Applications of Integration

Applications of Integration then explores volume calculations for solids of revolution using the disk and washer methods, as well as the shell method for cases where rotation is around an axis parallel to the region. Arc length is introduced, with integrals providing exact measurements for curves. Surface area of revolution formulas are derived and applied to both simple and complex shapes. Physical applications include work done by a variable force, with examples such as pumping liquids, lifting objects, and stretching springs using Hooke’s Law. The chapter also presents the concept of average value of a function and reviews its use in interpreting data. Special emphasis is placed on modeling and interpreting integrals in physics, engineering, and biology—for instance, calculating mass from density functions, determining center of mass and moments, and analyzing force distributions. By connecting integration to geometry, physics, and other sciences, the chapter demonstrates how the integral is a powerful unifying tool for solving diverse problems.