Applications of Integration Summary

Calculus 1 · Axiom Academy

SUMMARY Unit Summary: Applications of Integration Everything Unit 6 built from one idea — the definite integral as an accumulator: areas, volumes, lengths, averages, and net change. Every application here is the same move: slice the region into thin pieces, write the piece's contribution, and let the integral add them all up . Area between curves integrates the gap (top minus bottom) , never just one function — and the wider variable runs across the interval. Volumes of revolution pick a slice shape: a solid disk , a washer with a hole, or a cylindrical shell . Match the slice to the axis. Arc length and average value reuse the integral to measure a curve's length and a function's mean height. Net change is the Fundamental Theorem read forward: integrate a rate and you recover the total accumulated change. Core Concept Area Between Curves The area trapped between an upper curve f and a lower curve g is the integral of their vertical gap across [a,b] . When to use: a region bounded above and below by two curves. Watch out for: always top minus bottom ; if the curves cross, split the interval at each intersection so the gap stays positive. Core Concept Volume — Disk Method Rotate a region against the axis with no gap and each slice is a solid disk of radius R(x) ; the integral stacks their areas into a volume. When to use: the solid is gap-free against the axis of rotation. Watch out for: R(x) is the distance from the axis to the curve, not the raw f(x) unless the axis is y=0 .

This is the written version of the interactive lesson above. See the full Calculus 1 course.