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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 ranges over . Its area is therefore

Topics: Surface Of Revolution, Surface Wave, Surface Transportation Act...

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