# 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 [itex]x(t)[itex], [itex]y(t)[itex], with [itex]t[itex] ranging over some interval [itex][a,b][itex], and the axis of revolution is the [itex]y[itex] axis, then the area [itex]A[itex] is given by the integral

[itex] A = 2 \pi \int_a^b x(t) \ \sqrt{\left({dx \over dt}\right)^2 + \left({dy \over dt}\right)^2} \, dt [itex],

provided that [itex]x(t)[itex] is never negative. This formula is the calculus equivalent of Pappus's centroid theorem. The quantity

[itex]\left({dx \over dt}\right)^2 + \left({dy \over dx}\right)^2 [itex]

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 [itex][0,\pi][itex]. Its area is therefore

[itex]A = 2 \pi \int_0^\pi \sin(t) \sqrt{\left(\cos(t)\right)^2 + \left(\sin(t)\right)^2} \, dt = 2 \pi \int_0^\pi \sin(t) \, dt = 4\pi [itex].

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