gen-explain-volume-of-solids-of-revolution-as-you-would-to-a-cap-5d1e

Explain volume of solids of revolution as you would to a capable student, in 3-4 sentences.

When you rotate a 2D region around an axis (like spinning a curve around the x-axis), you create a 3D solid—imagine spinning a semicircle to form a sphere. To find the volume, we slice the solid into thin pieces perpendicular to the axis of rotation: **disk/washer method** uses circular cross-sections when slicing perpendicular to the rotation axis, while **shell method** uses cylindrical shells when slicing parallel to it. The key is setting up an integral that sums up these infinitesimally thin slices: for disks it's ∫π[r(x)]²dx, where r(x) is the radius from the axis to the curve. Choose whichever method gives you the simpler integral for your specific problem.