Wikipedia:Reference desk/Archives/Mathematics/2007 August 26

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August 26

World Mall - Yet again

Srry. OK let's try it this way. Let's pretend the octagon is automatically 64,000,000 square feet, and if the octagon is perfect in every way, how long would each side be?Jk31213 00:33, 26 August 2007 (UTC)

The sides of a regular planar octagon with area 64,000,000 are approximately 3640.718884498 long. Black Carrot 00:53, 26 August 2007 (UTC)

OK. Now how did u figure that out?Jk31213 00:57, 26 August 2007 (UTC)

Through (I assume) the formula for the area of a regular octagon: $A=2(1+{\sqrt {2}})s^{2}$. Strad 01:10, 26 August 2007 (UTC)

Which one can derive by thinking of a regular octagon as a square with triangles cut out of the corners, and using the formulas for the area of squares and triangles. J Elliot 03:08, 26 August 2007 (UTC)

You need to solve that formula for s:

$A=2(1+{\sqrt {2}})s^{2}$

$A/(2(1+{\sqrt {2}}))=s^{2}$

${\sqrt {A/(2(1+{\sqrt {2}}))}}=s$

Then plug in 64,000,000 for A and find s. StuRat 22:51, 26 August 2007 (UTC)

It's hard to know what to say about it that hasn't been said in response to one of your previous posts. Black Carrot 05:18, 26 August 2007 (UTC)

This question has been done to death. An octagon is just 8 identical triangles. The formula for the area of the triangle (when given the side length) is well known. So what seems to be the problem? 202.168.50.40 22:32, 26 August 2007 (UTC)

Decimals

What is a decimal point? Why we not read 6.23 six point two three not six point twenty three. Dharmend

See decimal. — Kieff | Talk 06:55, 26 August 2007 (UTC)

I read "6.23" as "Six Point Two Three". So what is the problem? 202.168.50.40 22:35, 26 August 2007 (UTC)

integral/antiderivatives..

I'm stuck on this integral:

((Integrate over a:0 to 2pi, b:-pi/2 to pi/2))

cos b (k - cos a cos b) da db / (k²+1 - 2k cos a cos b)^3/2

Anyone know how if possible? (not an advanced mathematician - may need links to post school techniques)

Please don't strain yourself on this one - it's not of great importance. I'm just curious if it's easily doable - specifically to get an 'easy' expression.. Maybe I missed something obvious? Thanks.87.102.44.85 14:02, 26 August 2007 (UTC)

Can we verify your intent? Do you mean the following?

$\int _{0}^{2\pi }\left(\int _{-\pi /2}^{\pi /2}{\frac {\cos(b)(k-\cos(a)\cos(b))}{\left(k^{2}+1-2k\cos(a)\cos(b)\right)^{3/2}}}\,db\right)da$

If so, I see nothing immediately obvious. Still, it may yield to a transformation of some sort. How does it arise? --KSmrq^T 14:42, 26 August 2007 (UTC)

erm yes (the integrals can be performed in either order - yes?)

It arises from an inverse square law 'force' at a distance from a spherical surface which has uniform 'density of things' on it.. I that makes sense. I usually make mistakes (but I have had a couple of gos at deriving this so hopefully it's ok)

The k is x/r where x is the distance from the sphere centre and r is the radius of the sphere. I've missed out a constant (Q/4pi) that doesn't affect the integration.

So I've said the quantity of stuff on the sphere at (r,b,a) spherical coordinates (I've used a=rotation around 'z', b = angle between point, origin and projection onto xy plain - not quite the same system as in above article) is Q r² cos(b) da db /4πr² (where Q is the total amount of 'stuff').

Then work out the distance from that point {defined by (r,b,a)} to distance x (and inverse square it), and finally multiply by the cosine of the angle a line from point on sphere (r,b,a) to x {eg (x,0,0)} makes to the line (0,0,0) to (x,0,0) (since the sine components cancel - eg eg like a force)...87.102.11.213 16:10, 26 August 2007 (UTC)

The sphere is centred on (0,0,0).87.102.11.213 16:16, 26 August 2007 (UTC)

I did not actually try this to see if it leads anywhere in this case, but in general, to find an antiderivative of f(cos x) with respect to x, you can try finding a primitive G(z) of g(z) = (1 − z²)^−1/2f(z), and check if ±G(cos x) works for you. If f was a rational or irrational function, then g is an irrational function, and with some luck you'll find it in List of integrals of irrational functions, or else you can perhaps massage it into the required form. This is the kind of thing that if it can be done, programs like Maple or Mathematica will probably do it for you.  --Lambiam 16:47, 26 August 2007 (UTC)

Thanks, I'm unfamilar with the method, (still working on it)

it did however lead me to another possibility..

replacing the 'cos x' with x giving a function I can integrate and using:

G(x) is the integral of g(x), G(cos x)= ʃg(x)sin(x)dx . I can integrate by parts, and can also work out ʃʃg(x) ʃʃʃg(x) etc giving

G(cos x) = sin(x)ʃg(x) - cos(x)ʃʃg(x) -sin(x)ʃʃʃg(x) .. etc

(that isn't exactly what you meant I think? or maybe), (still in progress)87.102.85.15 14:47, 27 August 2007 (UTC)

I don't think that's quite correct. If you take g(x) = 2x, you have G(x) = x². But G(cos x) = cos² x is not an antiderivative of g(x) sin x = 2x sin x. What I suggested is a straightforward application of integration by substitution (a.k.a. "change of variables"), with the correspondence between the variables given by z = cos x, so dz = −sin x dx = ±(1 − z²)^1/2dx. BTW, I made a mistake above where I wrote "(1 − z)^−1/2", which should have been "(1 − z²)^−1/2"; I've corrected that.  --Lambiam 16:27, 27 August 2007 (UTC)

Your right

G(cos x)= ʃg(x)sin(x)dx should have been

G(cos x)=-ʃg(cos x)sin(x)dx (AND I missed a minus sign too..) Oops. Back to beginning.87.102.85.15 17:23, 27 August 2007 (UTC)

Mathematica can evaluate the inner integral, with respect to either a or b, but the result takes two screens to display and involves nonelementary functions. So not much help there, at least not unless the integral can be simplified manually. Fredrik Johansson 16:52, 26 August 2007 (UTC)