Wikipedia:Reference desk/Archives/Mathematics/2018 May 4

= May 4 =

Calculating the centroid
A solid of revolution is produced by revolving y=sqrt(x), 0<y<6, around the y-axis. Find the y-coordinate of the centroid.

Since y^2=x, the integral of pi*y^5 from 0 to 6, divided by the integral of pi*y^4 with the same limits, returns y-bar = 5. This is marked wrong. How come? Imagine Reason (talk) 04:11, 4 May 2018 (UTC)
 * Because you are dealing with a solid of revolution, not a lamina. Actually the problem as stated does not give enough information because a solid of revolution is generated by revolving a plane region around an axis and you are only given a single curve.--Jasper Deng (talk) 05:50, 4 May 2018 (UTC)
 * Ugh, I wa only typing up the problem and missed the part where it's revolving (by 360 degs, just to be clear) the area enclosed by the curve and the y-axis, 0<x<36. 104.162.197.70 (talk) 10:53, 4 May 2018 (UTC)
 * I can't see anything wrong with what you did. Are you sure they wanted it revolved around the y axis rather than the x axis as that would give a more common shape? Dmcq (talk) 11:26, 4 May 2018 (UTC)
 * The question is a bit sloppy, but if we are looking for the centroid of a solid of revolution the answer looks correct. However if we are looking for the centroid of the surface generated, the answer is different.
 * More formally: under cartesian coordinates x,y,z, the solid defined by $$0\leq y \leq 6, \sqrt{x^2 + z^2} \leq y^2$$ does have its centroid at y=5 (and x=z=0). However the surface defined by $$0\leq y \leq 6, \sqrt{x^2 + z^2} = y^2$$ has its centroid at 24/5, a little below (intuitively, because the large y regions get a weight boost proportional to radius rather than to squared radius). Tigraan Click here to contact me 11:37, 4 May 2018 (UTC)
 * If we're talking about a surface we should probably be talking about one of constant thickness even if t is very thin. Dmcq (talk) 18:12, 4 May 2018 (UTC)
 * If the solid of revolution is defined as $$y^2\leq\sqrt{x^2+z^2} $$ (an infinite solid) then its centroid is at $$y=3$$. Ruslik_ Zero 14:01, 5 May 2018 (UTC)

Should this be corrected?
There's this page that states an x-axis parameterization as (ignoring constants) t*cot(t) here https://en.wikipedia.org/wiki/Quadratrix_of_Hippias, but then in another page, it says the same curve is -tcot(t)+it in the Branches and range section of here https://en.wikipedia.org/wiki/Lambert_W_function t*cot(t) =/=-t*cot(t)+it it has to be one or the other — Preceding unsigned comment added by 71.90.42.134 (talk) 07:31, 4 May 2018 (UTC)
 * No they are describing the same sort of thing but in different ways. One gives real coordinates (x(t), y(t)) where t is real whereas the other gives a complex number with real part x(t) and imaginary part y(t) with a complex number parameter t when the imaginary part is 0. Dmcq (talk) 11:43, 4 May 2018 (UTC)
 * [ec] I assume that, like you ignored the constants that just scale the shape, you understand that the minus sign does nothing but reflect the shape. So the apparent discrepancy is between $$t\cot t$$ and $$t\cot + it$$.
 * These two expressions refer to slightly different things. As you can see in the Quadratrix of Hippias article, the full description of a point on the curve, parameterized by t, is $$\gamma(t)=(t\cot t, t)$$. Whereas in the Lambert W function article, the point is $$w = t\cot t+it$$. As you can see, the expressions $$(t\cot t,t)$$ and $$t\cot t+it$$ are quite similar. In fact, if you take the complex number $$t\cot+it$$ and represent it as a point on the plane, that point is $$(t\cot t,t)$$. So it's the exact same thing, just that the first page is purely geometric, while the second considers the shape in the context of a complex function. -- Meni Rosenfeld (talk) 11:47, 4 May 2018 (UTC)

Balls into bins
If I place n balls into k bins. What's the probability that each bin but one contains at most $$n/\lfloor(k+1)\rfloor$$ of the balls? David Frid (talk) 16:13, 4 May 2018 (UTC)


 * Does your p have some relation to n and k, or is it arbitrary? -- ToE 18:25, 4 May 2018 (UTC)


 * I'm sorry... It was a typoDavid Frid (talk) 20:18, 4 May 2018 (UTC)


 * No worries. If I may continue questioning your question, is n/(k+1) really what you want?  Consider the specific case of n=61 balls placed into k=3 bins.  The most even distribution would be 21 balls in one bin and 20 balls in each of the other two.  You appear to be asking what the probability would be that, if the balls are randomly binned (with equal probability of a ball landing in any one of the bins, etc.), then only one bin would have an excess number of balls.  Let's get "excess" properly defined before we try for an answer.  Do you really want only one bin containing more than floor(n/(k+1))=15 balls?  Or did you want only one bin containing more than 20 balls? -- ToE 13:40, 5 May 2018 (UTC)


 * Assuming the question really is as asked for, this is extremely rare. Exactly how rare will depend on how big k and n are (absolutely and relatively), but for some context: in 100000 trials with 100 balls and 5 bins, only 5 met the criteria; in 20000 trials with 100 balls and 10 bins and then another 20000 trials with 20 bins, a big fat 0 met the criteria.  --JBL (talk) 23:07, 5 May 2018 (UTC)

I meant floor value, and I re-edited my question correspondingly 2.53.134.207 (talk) 03:46, 7 May 2018 (UTC)


 * The floor was never an issue, as "at most n/(k+1)" is the same as "at most ⌊n/(k+1)⌋" where integer values are concerned. (I consider your former wording more elegant, and used floor myself just to avoid saying 15.25 balls. Sorry for the confusion.  (Also, I just noticed that you floored the divisor, not the quotient. -- ToE 16:14, 7 May 2018 (UTC)) )  The question was whether you really wanted n/(k+1) (15 balls, in the example), as opposed to (n/k)+1 (21 balls) or n/k (20 balls).  (n/(k+1) just seemed sort of arbitrary for what I assumed to be your root question, but there is nothing wrong with it if that is what you truly intended.)
 * In any case, JBL's answer above may be the best you'll get here. If you let us know the range of n & k you are interested in, or explain the root question, then we might be able to provide a better approximation for the region of interest. -- ToE 08:17, 7 May 2018 (UTC)