In calculus, is it possible to create a volume or revolution by
revolving an area about a curved line, such as a parabola, rather than
an axis?  And if so, what would the accompanying equation be?

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Clarification of Question by floridaguy-ga on 08 Jun 2005 14:21 PDT

That should read "volume OF revolution" in the first line, not "volume
OR revolution."

If you built a real area (out of cardboard, say) and a real curve (out
of wire), you wouldn't be able to revolve the area around the curve
without bending something.

You might be able to define a way of doing it mathematically, but that
would be tricky.  Try drawing a curve and an area.  For each point on
the curve, you can draw a line perpendicular to that point.  You'd
want the parts where the line passes through the area to rotate around
that particular point on the curve.  However, it's fairly easy to draw
an example where the same bit of area would have to rotate around more
than one point, so it's not clear what you'd do there.

This is a free comment.

If your curves are f(x) and g(x), then make a small cylinder of height
dx, its radius is f(x) - g(x), thus integrate pi*(f(x) - g(x))^2 dx.

Floridaguy,
An interesting question. If you insist that the area to be rotated
remain undeformed, a rectangle, say, then obviously it can only be
rotated on a straight axis, i.e., two points on two sides of the
rectangle (or at two corners).
But since the question assumes that it can be rotated around a curved
line, presumably on an axis of the points where the curved line
intersects the  edges of the rectangle  (which is then also assuming
that these edges remain parallel during the rotation), as it rotates,
the rectangle will be deformed in two ways.

To try to visualize this, consider the form of the rectangle after it
has turned 90° from its initial position flat on a curved line in two
dimentions:
Now the "rectangle" will be curved like a piece of paper to lie in the line;
and the sides not intersected by the curve will themselves be curved
to allow the original point of contact between the "rectangle" and the
line to remain unchanged.  This deformation changes as the "rectangle"
rotates further, eventually returning to its original shape after
360°.

To visualize this it may help to recall beginner's differential
calculus and consider the rectangle as a mass of parallel lines
crossing the curved line, each of them rotating on the point where
they cross it, each defining a circle, so the skewing of the figure is
immaterial.

So where does this put us for calculating the volume of rotation?

I believe (?! Calculus is only a religion to them who know!) the shape
of the rotated figure will be defined by the rotation of the longer
portion of these parallel lines.  Depending on the placement of the
(simple) curved line, these could all be on one side of it, both sides
of it, or  - if the curve twice intersected the middle line
perpendicular to the plane of rotation -  this would define three
areas. The volume would be that calculated in the first instance by
the rotation of the larger area of the rectangle defined by the curved
line;
Or in the second and third instances by the sum of the rotations of
the two or three areas defined.

Can I write the formula?  NO.  
Am I right with my assumptions and description?  I don't know?

It seems to me like the formula would be more complicated than Euler-ga's, 
but then someone who chose that user name must know a lot more about
maths than I do, and maybe I shouldn't stuck my neck out.
Myoarin

Euler-ga's comment gives the formula for the volume between two curves
both being rotated around the x-axis. It does not relate to the
question.

Ref:-
Euler-ga's comment gives the formula for the volume between two curves
both being rotated around the x-axis. It does not relate to the
question.
I think the two curves both being rotated around the x-axis is Int
pi*f(x)^2 - Int pi*g(x)^2

I tend to agreee with euler-ga...

The rotation of f(x) about g(x) gives the curve g(x)-(f(x)-g(x)); when
g(x)=0 for the x-axis, the rotated curve becomes -f(x). Hence
euler-ga's answer.

Hi cliveac-ga,

This is not correct (but neither was I). g(x) - (f(x)-g(x)) =
2g(x)-f(x) is in fact a sort of pointwise reflection of f(x) in g(x)
(normally when you reflect in a line you reflect through the normal;
in this case we're reflecting through the horizontal at each point).

Euler-ga's formula seems to be derived from a similar idea, where you
rotate each point of f(x) around the point of g(x) with the same
x-coordinate. However, this is at serious variance with what is
normally meant by rotating around a given curve. For instance, take
f(x) = x/2 on the domain [0, 2]. To rotate this around the line y=x
we'd draw the reflection (in this case, the inverse function) f'(x) =
2x on [0, 1], and then join the two with circles perpendicular to the
3D vector (1,1,0). This gives us a cone with apex at the origin and a
base circle including points (2, 1, 0) and (1, 2, 0). The centre of
this circle is at (3/2, 3/2, 0) and the radius is therefore 1/sqrt(2).
The height of the cone is 3/sqrt(2) and the volume is therefore
(pi/3).(1/2).(3/sqrt(2)) = pi/(2.sqrt(2)).

Euler-ga's formula would instead give the second line as f'(x) = 3x/2
on [0, 2], and a total volume of 4.pi/3.

It all comes down to how exactly you want to define revolving an area
around a curve. The very notion of revolving something implies
revolving around a straight (and infinite) line, so it comes down to
which line we pick at any given point. We would like a solution that
corresponds to the normal definition of revolving around a line if the
curve in question happens to be a line.

Euler-ga has somewhat arbitrarily chosen to rotate around the line
y=g(x) at each point x. While this may seem sensible, bear in mind
that it would be equally arbitrary to revolve around x = g^{-1}(y) at
the same point, and this would give a completely different answer. If
we want to correspond with the usual meaning of revolving around a
line, we have to choose the normal (perpendicular) direction to
revolve in. However, as biophysicist-ga noted, this leads you into
cases where one piece of the area rotates around multiple points, or
none. I don't think this is necessarily a problem, unless you're
trying to do it with real (and not easily stretchable or deformable)
objects. As a mathematical process it should be doable, at least in
principle. In practice I'm sure you'd get some extremely hairy
integrals. Of course you can use numerical integration to get
approximate answers for specific problems, but I wouldn't hold out
much hope for general solutions except in extremely simple cases (e.g.
g(x) is piecewise linear).