Scheibenmethode

Scheibenmethode is the German term for the disk and washer methods used in integral calculus to compute volumes of solids of revolution. The method applies when a region in the plane is rotated around an axis, creating a solid whose cross‑sections perpendicular to the axis are either disks (solid circles) or washers (annuli). By integrating the area of these cross‑sections along the axis, one obtains the total volume.

For a rotation around the x‑axis, a function f(x) ≥ 0 defined on [a,b] generates a disk of radius f(x). The differential volume element is dV = π[f(x)]²dx. If a hole is present—i.e., if the region is bounded by both y = f(x) and y = g(x) with f(x) ≥ g(x) ≥ 0—the cross‑section is a washer with outer radius f(x) and inner radius g(x), giving dV = π([f(x)]² – [g(x)]²)dx. Analogous formulas hold for rotation around the y‑axis, with variable substitutions and differential elements swapped accordingly.

The Scheibenmethode is often presented as a systematic alternative to the method of cylindrical shells. While the shell method integrates parallel strips to the axis of rotation, the disk/washer method integrates perpendicular strips. Both yield equivalent results; the choice depends on which yields a simpler integrand. The method is taught in advanced high‑school and undergraduate calculus courses and is fundamental for solving problems involving solid geometry, physics (moment of inertia), and engineering. The technique generalizes to higher dimensions via the method of slices and to computational methods where numerical integration replaces analytic solutions.