A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around a straight line in its plane (the axis)^{[1]}.
Examples of surfaces generated by a straight line are cylindrical and conical surfaces when the line is coplanar with the axis, as well as hyperboloids of one sheet when the line is skew to the axis. A circle that is rotated about a diameter generates a sphere and if the circle is rotated about an coplanar axis other than the diameter it generates a torus.
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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 A is given by the integral
provided that x(t) is never negative. This formula is the calculus equivalent of Pappus's centroid theorem^{[2]}. The quantity
comes from the Pythagorean theorem and represents a small segment of the arc of the curve, as in the arc length formula. The quantity 2πx(t) is the path of (the centroid of) this small segment, as required by Pappus's theorem.
If the curve is described by the function y = f(x), a ≤ x ≤ b, then the integral becomes
for revolution around the xaxis, and
for revolution around the yaxis. These come from the above formula.
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,π]. Its area is therefore
For the case of the spherical curve with radius r, rotated about the xaxis
To generate a surface of revolution out of any 2dimensional scalar function y = f(x), simply make u the function's parameter, set the axis of rotation's function to simply u, then use v to rotate the function around the axis by setting the other two functions equal to f(u)sinv and f(u)cosv conversely. For example, to rotate a function y = f(x) around the xaxis starting from the top of the xzplane, parameterize it as for and .
Geodesics on a surface of revolution are governed by Clairaut's relation.
The use of surfaces of revolution is essential in many fields in physics and engineering. When certain objects are designed digitally, revolutions like these can be used to determine surface area without the use of measuring the length and radius of the object being designed.
