surface of revolution

A surface of revolution is a surface created by rotating a curve lying on some plane (the generatrix) around a straight line (the axis of revolution) that lies on the same plane. Examples of surfaces generated by a straight line are the cylindrical and conical surfaces. A circle generates a toroidal surface. If the curve is described by the functions

x(t)

y(t)

, with

t

ranging over some interval

a,b

, and the axis of revolution is the

y

axis, then the area

A

is given by the integral

A = 2 \pi \int_a^b x(t) \ \sqrt{\left({dx \over dt}\right)^2 + \left({dy \over dt}\right)^2} \, dt

provided that

x(t)

is never negative. This formula is the calculus equivalent of Pappus's centroid theorem. The quantity

\left({dx \over dt}\right)^2 + \left({dy \over dx}\right)^2

comes from the Pythagorean theorem. For example, the spherical surface with unit radius is generated by the curve x(t)=sin(t), y(t)=cos(t), when t ranges over

0,\pi

. Its area is therefore

A = 2 \pi \int_0^\pi \sin(t) \sqrt{\left(\cos(t)\right)^2 + \left(\sin(t)\right)^2} \, dt = 2 \pi \int_0^\pi \sin(t) \, dt = 4\pi

Surface Of Revolution

See also