Surfaces of revolution
A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation.
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 about any diameter generates a sphere of which it is then a great circle, and if the circle is rotated about an axis that does not intersect the interior of a circle, then it generates a torus which does not intersect itself (a ring torus).
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.
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 area Ay is given by the integral
provided that x(t) is never negative between the endpoints a and b. This formula is the calculus equivalent of Pappus's centroid theorem. The quantity
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