You can achieve this through IPOs after setting an animation for a surface , not a curve or anything else but a surface, along the choosen path
Placing yourself on frame 1 makes things clearer.
Once your animation is made, select the surface and set the “DupliFrames” button in the animation buttons (F5). A series of shadows will show you the eventual result of your extrusion. Create one loc/size/rot key for the surface. By manipulating the resulting IPOs you’ll be able to get all kind of results.
To make your end result real :
- select the surface and make the shawdows real by ctrl+shift+a > make dupli real.
- you have now a set of nurbs that you will join together by ctrl+j. This makes them one object.
- go in edit mode and select all vertices : then skin by pressing the FKEY once.
All this is very basic. There’s a slew of other parameters you can control to achieve all kinds of results and correct problems. Try this, don’t hesitate to ask questions as the power of the method entails complexity, hence the risk I omitted some essential detail or the chance that you’ll want to know more.