Maple 8 code:

with(plots): with(plottools): 
f := t -> t^3/2-2*t^2+2*t+1:
start := spacecurve([0,t,f(t)],t=0..3,thickness=3):
pic := n->
    cylinderplot(f(z),theta=0..n*2*Pi/30,z=0..3):
display(start,seq(rotate(pic(n),Pi/2,Pi/2,0),n=1..30),
    insequence=true,axes=normal,tickmarks=[0,0,0]);

1. This animation consists of 31 frames.  The first is a spacecurve drawn in the y-z plane, and the next thirty frames are increasingly larger segments of the corresponding solid of revolution, drawn using Maple's "cylinderplot" routine.
2. The viewpoint and color scheme were adjusted manually in Maple after the animation was constructed.
3. We used an animation editor to slow down the animation, and to display the first and last frames for a longer time than the intermediate ones.  This effect gives the viewer time to understand what is happening.
4. Change the definition of the function in the second line, to experiment with other curves and their solids of revolution.