Shell method (cylindrical shells)

Volume of revolution as nested cylindrical shells: $V=2\pi\int_a^b x\cdot h(x)\,dx$ .

The shell method finds a volume of revolution by summing THIN CYLINDRICAL SHELLS instead of disks or washers — each shell at distance $x$ from the axis has circumference $2\pi x$ , height $h(x)$ , and infinitesimal thickness $dx$ , giving $V=2\pi\int_a^b x\cdot h(x)\,dx$ . It's the natural choice when the disk/washer method would force an awkward solve-for- $x$ -in-terms-of- $y$ step.

Examples

Revolving the region under $y=x^2$ on $[0,2]$ about the $y$ -axis via shells gives $V=2\pi\int_0^2 x\cdot x^2\,dx=2\pi\int_0^2 x^3\,dx$ .