The Slicing Method

Calculus 2 · Axiom Academy

Finding volumes by integration: slice a solid into infinitesimally thin pieces, find each cross-sectional area, and sum them up. Imagine cutting a loaf of bread into thin slices. Each slice has some cross-sectional area A(x) and a small thickness dx , so its volume is about A(x)·dx . Add up every slice and, as the slices become infinitesimally thin, the sum becomes exact through integration. A(x) = area of the cross-section at position x · integrate from a to b along the axis 2. Circular Cross-Sections (Disk Method) Suppose every cross-section perpendicular to the x-axis is a circle of radius r(x) . A circle has area πr² , so A(x) = π[r(x)]². For example, rotate the curve about the x-axis from x = 0 to x = 4. Each slice is a circle of radius , so A(x) = π( )² = πx. Now build a solid whose base is the region under y = 4 − x² from x = −2 to x = 2, and whose cross-sections perpendicular to the x-axis are squares . A square of side s(x) has area s², and here the side spans the base, so s(x) = 4 − x². 4. Equilateral Triangle Cross-Sections Keep the same base ( y = 4 − x² , from x = −2 to x = 2), but now make each cross-section an equilateral triangle with its base on the xy-plane. An equilateral triangle of side s has area , and again s(x) = 4 − x². 5. Comparing Different Cross-Sections

This is the written version of the interactive lesson above. See the full Calculus 2 course.