Hollow Cylinders

Calculus 1 · Axiom Academy

When the spinning region doesn't touch the axis, every slice comes out with a hole — and the washer method handles it. Spin a region with a gap, and the solid is hollow Revolve a flat region around an axis and you sweep out a solid. But when the region floats above the axis instead of resting on it, the spin leaves an empty tube up the center — like a pipe, a bead, or a doughnut. Each slice through the solid is a washer : a disk with a circular hole punched out. Watch the sweep line cross the region between two curves. At every position it lays down one washer cross-section, and the running volume V … climbs as the washers accumulate — settling on the exact volume of the whole hollow solid. That accumulated stack of washers is the integral. The volume is just the washers, summed up — the running total the sweep line leaves stacked along the axis. Where R and r come from: the outer and inner curves Drag the slice anywhere across the region. The outer curve sets how far the washer reaches — its outer radius R(x) — and the inner curve sets the hole — its inner radius r(x) . The cross-section on the right redraws live, and its area is : the full disk minus the hole. Outer radius minus inner hole — every washer's area follows the two curves as you slide. Thinner washers, sharper answer

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