Surface of revolution

A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) one full revolution around an axis of rotation (normally not intersecting the generatrix, except at its endpoints).

The volume bounded by the surface created by this revolution is the solid of revolution.

Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on whether or not the line is parallel to the axis. A circle that is rotated around any diameter generates a sphere of which it is then a great circle, and if the circle is rotated around an axis that does not intersect the interior of a circle, then it generates a torus which does not intersect itself (a ring torus).

Properties

The sections of the surface of revolution made by planes through the axis are called meridional sections. Any meridional section can be considered to be the generatrix in the plane determined by it and the axis.

The sections of the surface of revolution made by planes that are perpendicular to the axis are circles.

Some special cases of hyperboloids (of either one or two sheets) and elliptic paraboloids are surfaces of revolution. These may be identified as those quadratic surfaces all of whose cross sections perpendicular to the axis are circular.

Area formula

If the curve is described by the parametric functions x(t), y(t), with t ranging over some interval [a,b], and the axis of revolution is the y-axis, then the surface area Ay is given by the integral

A

y

=

2

π

∫

a

b

x

(

t

)

(

d

x

d

t

)

2

+

(

d

y

d

t

)

2

d

t

,

{\displaystyle A_{y}=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 between the endpoints a and b.