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 rotation) that lies on the same plane.
Examples of surfaces generated by a straight line are the cylindrical and conical surfaces. A circle that is rotated about a (coplanar) axis through the center generates a sphere. If the axis is coplanar and outside the circle it generates a toroidal surface.
If the curve is described by the parametric functions , , with ranging over some interval , and the axis of revolution is the axis, then the area is given by the integral
provided that is never negative. This formula is the calculus equivalent of Pappus's centroid theorem. 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 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 x-axis, and
for revolution around the y-axis. 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 . Its area is therefore
For the case of the spherical curve with radius r, rotated about the x-axis
Rotating a function
To generate a surface of revolution out of any 2-dimensional scalar function
, simply make
the function's parameter, set the axis of rotation's function to simply
, then use
to rotate the function around the axis by setting the other two functions equal to
conversely. For example, to rotate a function
around the x-axis starting from the top of the
-plane, parameterize it as
Applications of surfaces of revolution
The use of surface of revolutions 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.