The Disk Method

Calculus 2 · Axiom Academy

Finding the volume of a solid of revolution by slicing it into thin circular disks whose radius is read straight off the curve. Take the region under a curve y=f(x) and spin it about the x -axis. Every vertical strip becomes a circular disk , and the disks stack edge to edge into a smooth solid. Watch the region below sweep all the way around — its outline is exactly the mirror of the curve, . Freeze a single representative strip at position x . Its height is the curve value f(x) , so when it spins it becomes a disk of radius r=f(x) . A disk is just a thin cylinder, so its volume is its circular area times its thickness. r=f(x) — read straight off the curve at that x . dx — an infinitesimally thin slice along the axis. — the area of the circular face. Add up every disk from x=a to x=b and the sum becomes the disk-method integral: 3. Visualizing the Cross-Sections Let's pin down a concrete curve: , rotated about the x -axis from x=0 to x=4 . As a slice sweeps left to right, each disk's radius is the curve's height there — , , up to — and the disks fit together with no gaps. Now add the disks up for on [0,4] . As more and thinner disks accumulate, their total volume settles on a single number — and the integral gives it exactly. You've seen how spinning a region builds a solid, why each disk's radius is just the curve's height, and how the integral adds the disks into a single volume. Scroll up to revisit any step.

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