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 , , 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. 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
:.
See also Solid of revolution Gabriel's Horn
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