math lessons - 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 $x (t)$ , $y (t)$ , 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

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

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

$\left({dx \over dt}\right)^2 + \left({dy \over dx}\right)^2$

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 $[0,π]$ . Its area is therefore

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

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