disk method

Finding the volume of a solid of revolution by slicing it into circular disks perpendicular to the axis of rotation.

The disk method finds the volume of a solid formed by revolving a region around an axis by integrating the area of circular cross-sections: $V=\pi\int_a^b [f(x)]^2\,dx$ , when the region touches the axis of revolution directly (no gap).

Examples

Revolving $y=\sqrt x$ on $[0,4]$ about the $x$ -axis gives $V=\pi\int_0^4 x\,dx = 8\pi$ .