revolvedsurface

Revolved surface, or surface of revolution, is a two-dimensional surface in three-dimensional space obtained by rotating a plane curve, called the generating curve, about a fixed line in the same plane, the axis of revolution. The surface is symmetric around that axis. If the generating curve is given by y = f(x) for x in [a,b], revolving about the x-axis yields the parametric description S(x, θ) = (x, f(x) cos θ, f(x) sin θ), with θ in [0, 2π). Equivalently, the implicit form is y^2 + z^2 = f(x)^2. The axis can be any line in the plane, producing the same family of surfaces after a coordinate change.

Common examples: a cylinder arises from f(x) = r; a cone from a linear f; a sphere from revolving a semicircle y = sqrt(R^2 − x^2) about the x-axis; a torus from revolving a circle of radius r whose center is offset from the axis by distance d (d > r).

Area and volume: for a curve y = f(x) revolved about the x-axis on [a,b], the surface area is 2π ∫_a^b f(x) sqrt(1 + (f′(x))^2) dx, and the enclosed volume is π ∫_a^b f(x)^2 dx. Similar formulas apply for other axes with appropriate substitutions.

Applications: surfaces of revolution appear in engineering, CAD, and computer graphics, where rotational symmetry simplifies modeling, surface integrals, and volume calculations.