Wikipedia:Reference desk/Archives/Mathematics/2012 February 24

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February 24

Solids of revolution by cylindrical shells

Resolved

 – I got 20π/3 which matches the computer answer, and it's due in an hour. Feel free to check my work. --Ks1stm (talk) ^{[alternative account of Ks0stm]} 17:24, 24 February 2012 (UTC)

My calculus textbook says "the volume of the solid obtained by rotating about the y-axis the region under the curve y = f(x) from a to b is V = ∫^b
_a 2 π x f(x) dx where 0 ≤ a < b." That's fine and all except one of my homework problems is to find the volume of the solid obtained by rotating the region between y = x² and y = 2 − x² about x = 1, and this region would produce an a less than 0. Will it still get me the correct answer if I attempt to find the volume by doing V = ∫¹
_-1 2 π (1−x) (2-2x²) dx or will it be thrown off by a being less than 0? Ks0stm ^{(T•C•G•E)} 00:55, 24 February 2012 (UTC)

Just impose a new coordinate system such that x=1 in the old one is x'=0 (y axis) in the new one and you won't have that problem. 24.92.85.35 (talk) 03:13, 24 February 2012 (UTC)

If I'm not mistaken, the definition is worded in a manner to ensure that the radii are all positive (and only included in the integration once, of course). In this case, because r=(1-x) and your limits are x=-1..1, these conditions still hold. --Kinu ^t/_c 03:26, 24 February 2012 (UTC)

Length of the curve of any polynomial

How is the length of any polynomial between two points calculated? --Melab±1 ☎ 04:24, 24 February 2012 (UTC)

See Arc_length#Finding_arc_lengths_by_integrating. StuRat (talk) 04:38, 24 February 2012 (UTC)

Would all polynomials substituted into the equation be integrable though? --Melab±1 ☎ 17:50, 24 February 2012 (UTC)

No, this problem is one of the motivations behind Elliptic integrals.--Salix (talk): 18:23, 24 February 2012 (UTC)

Clarification: all the integrands at arc length are integrable functions, in the sense that the definite integrals exist as finite real numbers (they also satisfy the conditions given by Riemann_integral#Integrability whenever f is a poynomial). However, there is no guarantee that the indefinite integral has an analytic expression, which leads us to Salix's comment above. (I only bring this up because, as I understand it, Melab is using 'integrable' to mean 'has an analytic antiderivative', which is not standard usage. (also feel free to chastise/correct me if I've missed something here;)) SemanticMantis (talk) 21:29, 24 February 2012 (UTC)

That is quite alright. But, I am confused if you mean that not all polynomials will produce an analytic antiderivative. I can understand that not all non-polynomial functions would have an analytical arc length. --Melab±1 ☎ 15:11, 25 February 2012 (UTC)

Yep, you have it. The antiderivative of a polynomial is a polynomial. But the integrands for arc length of a polynomial are not always polynomials, as Salix mentions. SemanticMantis (talk) 15:28, 25 February 2012 (UTC)

Question: limits of partially defined functions on the reals

I was alerted to a quirk about limits which I do not recall hearing before. Suppose f(x)=1 on the irrationals, and is undefined on the rationals. Topologically, when considered as a function from the irrationals into the reals, it has limit 1 as x approaches 0. It is appealing then to say that this function has a limit as a partially defined function on R. However the current version of the wiki article on function limits requires that f be defined at least on a (connected) open interval with endpoint 0. This stricter requirement would exclude the function described above from having a limit, due to the dense set of removable discontinuities.

• Question(s): Is there anything foundationally wrong with taking the wiki definition and "excusing" countably many removable discontinuities? (I.e. we would say "if x is less than delta from c AND f(x) is defined, then f(x) is less than epsilon from L") Does anybody know if any authors address this?

I'm not interested in altering the wiki article, this is just curiosity. Rschwieb (talk) 16:35, 24 February 2012 (UTC)

You could say that the property holds almost everywhere, if nothing else. Looie496 (talk) 18:18, 24 February 2012 (UTC)

That doesn't work. Consider the function that is 1 on the irrationals and 0 on the rationals instead of undefined. The limit of this function at 0 doesn't exist. The function you describe is actually continuous (the pre-image of open sets is open), but its domain is the irrational numbers and so we're dealing with the topology of the irrationals. There's no sense in which a function can be undefined at a point in its domain. Rckrone (talk) 18:29, 24 February 2012 (UTC)

Limit of a function#Functions on metric spaces appears to comfortably accommodate this (including sparse/countable domains); it involves only defining the set (domain) over which the limit is taken (as opposed to allowing a domain on which a function is undefined(!)), and that there must exist elements of the domain within any nonzero distance from the limiting point not counting the limiting point itself. — Quondum^☏✎ 18:35, 24 February 2012 (UTC)

(edit conflict) What you're talking about makes perfect sense. It's not usually discussed in elementary calculus texts because it's not necessary for talking about derivatives.

If f is a real-valued function defined on a subset E of R, then it makes sense to speak of the limit of f at a number a whenever a is a limit point of E. This means that every open interval containing a, no matter how small, contains a point of E other than a itself. (In fact, limits will still make sense if you drop the "other than a itself" part, in which case the notion is that of an adherent point.) It is not necessary that the set of points where f is undefined be countable, though this is a sufficient condition to ensure that every point is a limit point.

These kinds of limits will be addressed, at least implicitly, in any mathematical analysis textbook that discusses topology (namely, metric spaces or topological spaces). But this subsumes ordinary limits within a much larger theory and requires a degree of abstraction that goes far beyond what is needed for our purposes. The mathematical analysis textbook by Zorich discusses the more general concept of a limit we've been talking about well before it reaches topology. I can't give you a page number because I don't have the English version.96.46.204.126 (talk) 18:41, 24 February 2012 (UTC)

Haha, well if it's not in English then don't worry about it. I'm glad to hear my intuition was OK. Historical context: this all arose when I discovered a user "improved" l'Hopital's Rule by showing the g'(x) nonzero hypothesis was superfluous. All the analysis texts on my shelf used that hypothesis, so I was doubtful this person had outsmarted 20 generations of mathematicians. Just recently I found one reference which uses the same logic as that user. (First line of proof: "since lim f'/g' exists, g' does not vanish on a connected interval with endpoint a"). Is this not just a misconception? Maybe it's not so much of a misconception as an overstrong definition of limit. If one demands that in the definition of a limit that the function has to be defined on an interval adjacent to the limit point, then yes, the g'(x) nonzero condition is superfluous. Rschwieb (talk) 19:20, 24 February 2012 (UTC) Update: Checked 10 analysis books on a shelf. One of the authors (Kenneth Rogers) uses the proof I find questionable. S.G. Krantz assumes the two functions to be differentiable on an entire neighborhood including c, which I think probably does imply the g'(x) nonzero hypothesis. The other eight (including Rudin and Stewart) use the g'(x) nonzero hypothesis. Rschwieb (talk) 19:53, 24 February 2012 (UTC)

No, sorry, I meant that there is an English translation of it, but I don't have the translation, so I can't give you a page number in the English version. In Theorem 5.5.1 of the book you linked to, most mathematicians would include the hypothesis g′(x) ≠ 0 over some interval rather than relying on the technicalities of their particular definitions. It's likely that these technicalities are made clear wherever he defines limits. By the way, I can see no reason that the set of points at which g'(x) = 0 should be countable. 96.46.204.126 (talk) 19:59, 24 February 2012 (UTC)

I think that if you allow g′(x) to be zero for some x and choose to say nothing about what f′(x) does at those points, then L'Hopital's rule becomes false. For example, let f′ and g′ both be continuous, and let g′ be identically zero on infinitely many intervals approaching a, but nonzero elsewhere. Make f′ identical to g′ except on the intervals where g′ is zero, where you let f′ go crazy (while still tending to zero at a). You'll be able to pick f′ in such a way that L'Hopital's rule fails. (f and g will be the corresponding antiderivatives taking the value 0 at a. For example, f(x) will be the integral of f′ from a to x.) 96.46.204.126 (talk) 20:28, 24 February 2012 (UTC) [Correction: This may only work if you let g′ spend enough time being 0. 96.46.204.126 (talk) 20:37, 24 February 2012 (UTC)]

I think some careful thought is needed for this one. Just because it is possible to define a limit over a subset of R doesn't mean that the limit over R is defined, and in l'Hôpital's rule we are working over R. If we allowed such an "excusing" (even on a countable subset), we would be able to find limits in contexts where they do not exist. I also think it is possible that the apparent leap in the proof (of theorem 5.5.2) from the existence of a limit to a the nonzero statement may be tricky enough and uninteresting enough that it simply has not tackled adequately in the literature. Surely if the nonzero requirement was both interesting and necessary, there should be a reference that shows that it is necessary? It is not at all clear to me that the leap is false. I would guess that it may be possible to show that the existence of a limit implies the existence of a neighborhood on which the function (of which the limit is being taken) is defined. — Quondum^☏✎ 05:30, 25 February 2012 (UTC)

It pivots on what you define as "limit". If one insists that limited functions are defined on a connected interval approaching c, then by definition when you say lim f'/g' exists, you can infer g'(x) nonzero on said interval. This enhanced definition of limit implies the g'(x) nonzero hypothesis. If one uses the more general definition of existence of limit, then the example I gave on the l'Hopital's talk page shows that there exist functions f, g such that f/g has a limit 0 as x approaches 0, but f/g is undefined at many points in every interval (0,b). Thus in comparison, the g'(x) nonzero hypothesis is the more general of the two. Strengthening the definition of limit is only useful for simplicity. I do not recall ever hearing this enhanced definition in my career, so that's why I'm now wary of it. Rschwieb (talk) 19:08, 27 February 2012 (UTC)

And I agree with you in terms of what is to be presented; changes are new and non-standard. On a tangent, it is a little discomforting that there is no obviously robust rule for when l'Hôpital's rule applies; all of them seem to have twiddles to exclude pathological cases, or to be rather sensitive to details. But this is by the by; this kind of thing is also not too unusual I guess. — Quondum^☏✎ 19:28, 27 February 2012 (UTC)

Products

A product can be expressed in terms of factorials;

$\prod _{k=0}^{n-1}(N-k)={\frac {N!}{(N-n)!}}$

Can a product of every other number be similarly expressed?

$\prod _{k=0}^{n-1}(N-2k)$

Thanks, SpinningSpark 19:00, 24 February 2012 (UTC)

If N is even, then we have something like

$\prod _{k=0}^{n-1}(N-2k)={\frac {N!2^{N/2-n}(N/2-n)!}{(N-2n)!2^{N/2}(N/2)!}}.$

A similar expression is available if N is odd. Sławomir Biały (talk) 19:33, 24 February 2012 (UTC)

Beautiful! <3 --COVIZAPIBETEFOKY (talk) 20:09, 24 February 2012 (UTC)

Thanks for that. That's what I was afraid of, it has to be divided into odd and even expressions. SpinningSpark 21:22, 24 February 2012 (UTC)

Actually, that does not work even for even N. For instance,

8 × 6 × 4 = 192

has N = 8 and n = 3. Feeding those values into the expression I make 105 which is incorrect. SpinningSpark 12:23, 25 February 2012 (UTC)

Sorry, I was giving the product of the odds rather than the evens. The correct expression is

$\prod _{k=0}^{n-1}(N-2k)={\frac {2^{N/2}(N/2)!}{2^{N/2-n}(N/2-n)!}}$

for N even. For N odd it's

$\prod _{k=0}^{n-1}(N-2k)={\frac {N!2^{(N-1)/2-n}((N-1)/2-n)!}{(N-2n)!2^{(N-1)/2}((N-1)/2)!}}.$

Sławomir Biały (talk) 12:43, 25 February 2012 (UTC)

Still not working for the even case. The test example above is not even returning an integer now. The odd case seems ok though, at least for N=9. SpinningSpark 14:47, 25 February 2012 (UTC)

misprint fixed. It should work now. Sławomir Biały (talk) 15:09, 25 February 2012 (UTC)

Simplified, the formula for even N is

$\prod _{k=0}^{n-1}(N-2k)={\frac {2^{n}(N/2)!}{(N/2-n)!}}$.

If you want the same expression to work for both even and odd N, then you can use the Gamma function:

$\prod _{k=0}^{n-1}(N-2k)={\frac {2^{n}\Gamma (N/2+1)}{\Gamma (N/2-n+1)}}$. 98.248.42.252 (talk) 15:27, 25 February 2012 (UTC)

Ah...gamma function. Thanks. SpinningSpark 16:14, 25 February 2012 (UTC)

You've got the formulas now, but this being the reference desk, let me point to a reference. Factorial#Double_factorial tells about the product of every other number starting from 1 or 2, and how to write this in close form using the factorial function. Dividing two of these gives you the product you want. – b_jonas 17:16, 25 February 2012 (UTC)