Volumes by Cross-Sections

AP Calculus · Axiom Academy

LESSON Volumes by Cross-Sections Finding volumes using known cross-sectional shapes Step 1: Volume as an Integral of Cross-Sectional Area We already know how to find volumes of solids of revolution using discs and washers, where every cross-section is a circle. But what if the cross-sections are squares, triangles, or semicircles instead? The general principle is the same: slice the solid perpendicular to an axis, find the area of each slice, and integrate. where A(x) is the area of the cross-section at position x, perpendicular to the x-axis. Suppose a solid has a base in the xy-plane bounded by two curves, y = f(x) on top and y = g(x) on the bottom. Cross-sections perpendicular to the x-axis are squares . Each square has a side length equal to the vertical distance between the curves: Step 3: Other Cross-Section Shapes The same idea works for any shape. If the side (or diameter) of the cross-section is s = f(x) − g(x), here are the most common formulas: Step 4: Example — Square Cross-Sections Problem: A solid has a base bounded by and from x = 0 to x = 4. Cross-sections perpendicular to the x-axis are squares. Find the volume. Step 2 — Cross-sectional area: Step 5: Example — Semicircular Cross-Sections Problem: Same base region ( , , from 0 to 4), but now the cross-sections perpendicular to the x-axis are semicircles with diameter equal to the base width. Step 3 — Cross-sectional area:

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