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