Surface of revolution of 3D curve

How do you calculate area of surface of revolution of 3D curve? Though using integration I am able to calculate the area, but I would like to know a simpler method such as Pappus theorem for 2D curve.

Is the Pappus theorem limited to 2D curve or is the generalisation of Pappus theorem for 3D curve available?

I would also like to know that is there a theorem which says that the line about which surface of revolution of a given curve has minimum area should pass through the centroid of the curve?

[John D'Errico]

Please define what you mean by a 3D curve. Many people lately seem to be calling a surface a curve. Also define what the revolution is about. An axis? Some line? Something else? Without these informations, your question is unanswerable.

[Andrew Newell]

A MATLAB forum doesn't seem like the right place to ask this question. You need a math forum. Try browsing through http://archives.math.utk.edu/news.html.

Answer 1

[Amit Kumar]

I think this will help you a bit its the theorem you asked for and just try to make the answer out of it. The first theorem of Pappus states that the surface area of a surface of revolution generated by the revolution of a curve about an external axis is equal to the product of the arc length of the generating curve and the distance traveled by the curve's geometric centroid.

Similarly, the second theorem of Pappus states that the volume of a solid of revolution generated by the revolution of a lamina about an external axis is equal to the product of the area of the lamina and the distance traveled by the lamina's geometric centroid.