Chapter 8: Further Applications of Integration

Surface area of revolution is revisited and extended, showing how to compute the exact area generated by rotating curves around an axis. The chapter then moves into applications involving fluid pressure and force, analyzing hydrostatic pressure on vertical and horizontal surfaces with integrals accounting for varying depth and density. Probability density functions (PDFs) are introduced as a way to model continuous random variables, connecting integration to probability theory by computing cumulative distribution functions (CDFs) and probabilities over intervals. Parametric equations are explored for modeling motion and curves, with derivatives and integrals applied to compute velocity, speed, arc length, and area under curves in parametric form. Polar coordinates are introduced with their unique graphing techniques, and integration in polar form is used to find areas of sectors and regions bounded by polar curves. The chapter concludes by uniting these tools in modeling complex, real-world systems—from calculating forces on submerged structures to finding areas enclosed by intricate curves—highlighting the versatility and power of integration beyond standard geometric problems.