My Calc 1 class is going to be making models of the 3-dimensional shapes produced when curves are rotated about the x-axis as part of a unit on finding the volumes of such shapes.

I plan to model the shape produced by y=x^(1/2) starting from x=0 to whatever upper boundary is the square of the radius of the material I find. I will need to make a template scaled such that one graphical unit equals some measurement, careful to make the scale small enough that the curve is easily recognizable. Afterwards, I'll make a mold of the shape that can be filled with water to validate our calculation of its volume.

I have a lathe but it is not quite operational yet. If I can get it working I'll make the shape out of wood. If not, I'll have to mount some softer material in my drill press and turn it down like Technician did with his QDV pistons.

My first question is: what material should I use and how should I mount it to a 1/2" chuck? Styrofoam was the first thing to come to mind and I considered drilling a hole partially through a piece and gluing a dowel into it. I have a feeling that one slip with sandpaper will cause the dowel to shear right out of the hole or at least damage the foam. A heavier material wouldn't be as fragile, but it would be harder to shape into a curve using a drill press.

My second question is: What curves do French curves actually contain? It would be convenient to use a French curve to gauge the shape of my workpiece as I already have several, but I don't know the equations (if any) of the curves they contain. Other than sitting down with my calipers to make an x&y table to figure out the equations, I have no way of knowing. The internet hasn't been very helpful either. (And the fact that there are many types...)

Lastly, would I be wasting my time trying to make a shape like this? Are there any ready-made cones produced by specific equations that I can just buy? I know rocket nose cones come in all different varieties, especially parabolic.

All input is appreciated as always.

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