Topic 5 of 6
Applications of Integration
Translate geometry and physical accumulation into integrals with units, bounds, and meaning intact.
01Decision guideWasher method or shell method?Choose slices by geometry: perpendicular slices create washers, parallel slices create cylindrical shells.02Method guideArea between curves without guessing top and bottomFind intersections, choose a slice direction, and integrate the positive geometric difference across each interval.03Concept explainerWhere the arc-length formula comes fromApproximate a smooth curve by tiny line segments, then let the partition refine until the polygonal lengths converge.04Concept explainerAverage value of a function on an intervalTotal accumulated output divided by interval length—the continuous counterpart of an arithmetic mean.05Method guideSetting up work and fluid-force integralsSlice the changing force into small contributions, express every dimension in one variable, and integrate with units attached.