From: "B.J. Mares"
Newsgroups: sci.math
Subject: Volumes of rotation calculus problem
Date: Tue, 20 May 1997 20:23:19 -0700
I am trying to find the conditions on f(x) that satisfy the following
equation.
DefiniteIntegral( f(x)*(f(x)-2x), x = x1 to x2 ) = 0
If the integral is indefinite, it is obvious that the only two solutions
are f(x)=0 and f(x)=2x. When limits of integration are established,
several other functions satisfy the equation. It seems there must be
some differential equation that describes f(x).
I imagine if you are reading this, you are probably wondering what
purpose this equation has. When that equation is satisfied, the volume
generated by rotating the region defined by the function in the
specified interval about the x-axis is equal to the volume of revolving
it about the y-axis.
I don't have any idea how to approach this problem with my current
mathematical knowledge. I suspect maybe the methods of calculus of
variations might be helpful, but I don't know that much about that.
Thank you in advance.
B.J. Mares
==============================================================================
From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Volumes of rotation calculus problem
Date: 21 May 1997 17:41:43 GMT
In article <33826AA7.4D22@teleport.com>,
B.J. Mares wrote:
>I am trying to find the conditions on f(x) that satisfy the following
>equation.
>
>DefiniteIntegral( f(x)*(f(x)-2x), x = x1 to x2 ) = 0
>
>If the integral is indefinite, it is obvious that the only two solutions
>are f(x)=0 and f(x)=2x.
(You must be thinking of continuous functions, say, since there are
other solutions in this case too: e.g. f(x) =
0, if x1 <= x < (x1+x2)/2,
2x, if (x1+x2)/2 <= x < x2. )
This question is interesting because of the mathematics to which it
will introduce you.
Your defining property may be written Int( (f(x)-x)^2 ) = Int( x^2 ),
which is to say f(x) = x + g(x), where g is any (continuous,say) function
with Int( g^2 ) equal to a certain constant (depending on x1 and x2).
There are many such functions g, but what's more interesting is to
see how they are related geometrically. For any two continuous functions
F and G defined on the interval [x1, x2] one can define the
"inner product" = Int( F(x)*G(x) ) . (An inner product is a
function which shares the basic characteristics of the dot product in
Euclidean space.) In particular, the "length" of a function may then be
defined by || F || = Sqrt( ) which in this case is Int( F(x)^2 ).
Thus the set of functions g in the previous paragraph may be described
as the "sphere" of a certain radius in this set of functions!
If v1, v2, ..., v_n are perpendicular vectors of length 1 in
Euclidean space, then the unit sphere is the set of vectors of the form
Sum( t_i v_i ) such that the sum of the squares of the numbers t_i is 1.
The sphere of radius R is then found by scaling. Well, precisely the
same construction can be used in your case, although we'll have to
finesse the fact that the space is infinite-dimensional. What you need
is a collection of functions to take the place of the v_i. This would be
any (complete) set of functions F_i each of length 1 and each perpendicular
to all the others (in the sense of the previous paragraph).
For example, if x1=1 and x2=2, then the functions
F1(x) = 1 (constant)
F2(x) = sqrt(3)*(2*x-3) (linear)
F3(x) = sqrt(5)*(6x^2-18x+13) (quadratic)
"etc." forms such a family of functions. You can then form solutions to
your original problem of the form
f(x) = x + (t1*F1 + t2*F2 + t3*F3 + ... )
where the t_i are any real numbers satisfying (7/3) = t1^2 + t2^2 + t3^2 +...
(The two solutions you proposed correspond to t1 = +-(3/2), t2=t1/sqrt(27),
other t_i = 0; but as you can see there are infinitely many other
solutions even just among the linear polynomials.)
Discussions of general inner products are in most linear algebra books;
in particular, you'll want the Gram-Schmidt process for finding these
complete orthonormal sets. Particular examples of these are known as
orthogonal polynomials (of which there are some famous sets) and are
used in Fourier analysis as well. More generally, you are seeing here the
tip of functional analysis, which is what is needed if you need to make
precise just what you mean by "function", and if you hope to do a
careful job of understanding these possibly infinite sums.
dave