Volumes of Revolution

Building Solids from Slices

Integration extends from area to volume. A solid of revolution is formed by rotating a plane region about an axis — spinning a curve to sweep out a 3D shape (a sphere from a semicircle, a cone from a line). Its volume is found by slicing it into thin pieces whose volumes integrate up, exactly as areas were built from thin strips.

The disk method slices perpendicular to the axis of rotation. Each slice is a thin disk of radius $r(x)$ (the distance from the axis to the curve) and thickness $dx$, with volume $\pi r(x)^2\, dx$. Integrating gives $$V = \int_a^b \pi\, [r(x)]^2\, dx.$$ For a region rotated about the $x$-axis with $r(x) = f(x)$, this is $\int_a^b \pi f(x)^2\, dx$. If the region has a hole (rotated with a gap from the axis), the washer method subtracts the inner disk: $\pi(R^2 - r^2)$.

The shell method slices parallel to the axis, into thin cylindrical shells of radius $x$, height $f(x)$, and thickness $dx$. Unrolling a shell gives volume $2\pi x\, f(x)\, dx$, so $$V = \int_a^b 2\pi x\, f(x)\, dx.$$ Disks and shells are two ways to slice the same solid; often one is far easier to set up than the other depending on the axis and the region.

The unifying idea — slice, approximate each slice's volume, integrate — is the same principle as Riemann sums, now in three dimensions. It also computes arc length, surface area, work, and center of mass: any quantity that accumulates over an extent is an integral of its infinitesimal pieces.

Common pitfall: forgetting to square the radius in the disk method, or confusing the disk and shell setups. A disk's cross-sectional area is $\pi r^2$ (area of a circle), so its volume element is $\pi [r(x)]^2\, dx$ — the radius is squared. And disks slice perpendicular to the axis while shells slice parallel; mixing up which method matches the geometry (or dropping the $2\pi x$ factor in shells) produces the wrong integral.