Surface of revolution

Lore & Background

A surface of revolution is formed by rotating a curve, called the generatrix, one full revolution around an axis of rotation. The axis normally does not intersect the generatrix except at its endpoints. The volume bounded by this surface is known as the solid of revolution. Examples include cylindrical and conical surfaces, generated by rotating a straight line that is either parallel to the axis or not, respectively. A circle rotated around any diameter generates a sphere, of which the circle becomes a great circle. If the circle is rotated around an axis that does not intersect its interior, it generates a torus that does not intersect itself, specifically a ring torus.

Reader's Guide

The sections of a surface of revolution made by planes through the axis are called meridional sections, and any such section can be considered the generatrix in the plane determined by it and the axis. Sections made by planes perpendicular to the axis are circles. Some special cases of hyperboloids (of one or two sheets) and elliptic paraboloids are surfaces of revolution, identifiable as quadratic surfaces whose cross sections perpendicular to the axis are circular. The surface area can be computed using integral formulas derived from Pappus's centroid theorem, involving parametric or explicit functions. For a curve described by parametric functions x(t), y(t) rotated around the y-axis, the area is given by an integral of 2π times x(t) times the arc length element. Similar formulas apply for rotation around the x-axis or when the curve is given as y = f(x). These formulas are the calculus equivalent of Pappus's theorem, where the quantity 2πx(t) represents the path of the centroid of a small segment of the curve.

Did You Know?

• A surface of revolution is created by rotating a curve (the generatrix) one full revolution around an axis of rotation.
• Examples of surfaces of revolution include cylindrical surfaces, conical surfaces, spheres, and ring tori.
• The sections of a surface of revolution made by planes perpendicular to the axis are circles.
• The surface area formula for revolution around the y-axis is the calculus equivalent of Pappus's centroid theorem.