The Disk Method

Calculus 1 · Axiom Academy

Spin a region around an axis, slice the solid into thin circular disks, and add them up with an integral. Take the region bounded by a curve y=f(x) , the x -axis, and the lines x=a and x=b . Spin that flat region a full turn around the x -axis and it traces out a three-dimensional solid of revolution . Press play to watch the region rotate and the solid appear. Slice the solid with a plane perpendicular to the axis. Because every point of the region was swept in a full circle, that cross-section is a solid circular disk . Its radius is just the height of the curve at that x , and it has a tiny thickness dx . Watch the slicing plane move and read the disk's volume update. R(x)=f(x) — the distance from the axis out to the curve. dx — an infinitesimally thin slab along the axis. — the area of the circular face, since area . 3. Stack the Disks and Integrate Now line the disks up side by side from x=a to x=b . Their combined volume approximates the solid — and the thinner you slice, the better it fits. That sum of is a Riemann sum ; as it becomes a definite integral. R(x) — radius function (axis to curve) [a,b] — interval the solid spans Worked example: rotate on [0,4] The animation stacks disks under from x=0 to x=4 . Follow the radius all the way through: 2. Square it (the step everyone forgets): . You've seen a region revolve into a solid, isolated a single disk of radius R(x) , and stacked disks into the integral . Scroll up to revisit any step.

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