Applications of Integration Summary

Calculus 2 · Axiom Academy

SUMMARY Applications of Integration Summary How integration becomes the universal tool for accumulation — adding up infinitely many infinitesimal pieces to find areas, volumes, lengths, and physical quantities. Integration is accumulation. Every application follows one principle: sum infinitely many infinitesimal contributions over an interval. Geometry comes first. Even physics applications rest on geometric reasoning to set up the element being integrated. Many problems, multiple routes. Vertical vs. horizontal strips, disk vs. shell — choose whatever makes the algebra simplest. Visualization beats memorization. Understanding the picture or physical setup matters more than recalling a formula. Real-world reach. These techniques model architecture, manufacturing, fluid dynamics, and centers of gravity. Core Concept Area Between Curves Sum the heights of thin vertical strips between an upper curve and a lower curve. When the region is easier in terms of y , integrate horizontal strips instead: . When to use: the bounds come from where the curves intersect; split the integral if they cross inside the region. Watch out for: always verify which curve is on top (or on the right) over the interval — don't assume from the equations alone. Core Concept Volumes by Slicing Volume is the integral of the cross-sectional area. Place an axis perpendicular to the slices, then express each slice's area — square, rectangle, semicircle, or equilateral triangle — as a function of x or y .

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