Washer Cross Sections

Calculus 2 · Axiom Academy

When a rotated region doesn't touch the axis, every cross section is a ring — and its area, integrated, is the volume. A washer is a flat disk of outer radius R with a concentric hole of radius r removed — what geometry calls an annulus . Watch its area assemble: fill the full outer disk , then punch out the inner disk . What survives is the ring. Take a region trapped between two curves that both sit above the axis. At a position x , the slice runs from the lower curve up to the upper curve. Spin that slice around the axis: the far edge sweeps a circle of radius R(x) , the near edge sweeps a circle of radius r(x) , and the gap between them is the ring. The radii are exactly the distances from the axis to each curve at that x . Distance from the axis up to the outer (farther) curve. Distance from the axis up to the inner (nearer) curve. The whole game is reading off which curve is farther from the axis (that gives R ) and which is nearer (that gives r ). Everything after is plugging in. 3. Stack the Washers, Integrate One washer of thickness dx has volume . Sweep across the region from x=a to x=b , dropping a washer at every position, and add them all up. That sum is the integral — watch the running total climb as the sweep advances. Sketch the region and mark the axis of rotation. R(x) = distance from axis to the outer curve. r(x) = distance from axis to the inner curve. Find the limits a, b (where the curves meet).

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