Surface of revolution
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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 <math>x(t)<math>, <math>y(t)<math>, with <math>t<math> ranging over some interval <math>[a,b]<math>, and the axis of revolution is the <math>y<math> axis, then the area <math>A<math> is given by the integral
- <math> A = 2 \pi \int_a^b x(t) \ \sqrt{\left({dx \over dt}\right)^2 + \left({dy \over dt}\right)^2} \, dt <math>,
provided that <math>x(t)<math> is never negative. This formula is the calculus equivalent of Pappus's centroid theorem. The quantity
- <math>\left({dx \over dt}\right)^2 + \left({dy \over dx}\right)^2 <math>
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 <math>[0,\pi]<math>. Its area is therefore
- <math>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 <math>.