22/7 - pi integral and a sci.math challenge
Dave L. Renfro
is 22/7 - pi:
The integral from x = 0 to x = 1 of
(x^4)*(1 - x)^4 / (1 + x^2)
is equal to 22/7 - pi.
The evaluation by hand is much simpler than it looks,
or at least it's much simpler than I thought it would
be when I once decided to see how hard it'd be to do
it by hand. For those interested, the details are posted
here:
ap-calculus -- something for pi day (11 March 2008)
http://mathforum.org/kb/message.jspa?messageID=6133376
See also:
http://en.wikipedia.org/wiki/Proof_that_22/7_exceeds_%CF%80
The other day I came across Problem #557 in "Pi Mu Epsilon
Journal" [Volume 7, Number 9, Fall 1983, p. 615], which is:
"It is known and easy to show with elementary calculus that
[the integral above] = 22/7 - pi. Find a definite integral
whose value is 193/71 - e, where e is the base of natural
logarithms."
I've since come across some solutions [Vol. 8, No. 1,
Fall 1984, pp. 59-60], but all 5 published solutions
are extremely contrived:
(a) integrate 1 from x=0 to x = 193/71 - e
(b) integrate 193/71 - e from x=0 to x=1
(c) integrate 122/71 - e^x from x=0 to x=1
(d) integrate e^x from x=0 to x = ln(193/71)
(e) integrate 122x^70 - e^x from x=0 to x=1
At the end is the following editorial note:
"It was hoped that some delightful integral such as
the given one for 22/7 - pi would be found. Perhaps
some clever reader will still discover an elegant
integral for the desired 193/71 - e."
Thus far I have only gotten up to the 1986 Pi Mu
Epsilon Journal issues, so maybe at some later
time something "delightful" was published. However,
I thought I'd throw this out to the sci.math community
and see what you people come up with.
Dave L. Renfro
David C Ullrich
> There is a neat folklore definite integral whose value is 22/7 - pi:
>
> The integral from x = 0 to x = 1 of
>
> (x^4)*(1 - x)^4 / (1 + x^2)
>
> is equal to 22/7 - pi.
Holy cow, this means that pi = 22/7 after all!
(The integral vanishes by Cauchy's Theorem, since of
course the integrand is analytic...)
Han de Bruijn
Of course not. The integrand is positive, the integration interval is
nonzero and this is Real Analysis.
Han de Bruijn
David C. Ullrich
Erm, thanks. See, actually it was a joke.
You really thought I ... never mind.
>Han de Bruijn
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
master1729
Ullrich , plz go read a math book !!
Seriously !
sigh ...
regards
tommy1729
Ray Vickson
You are a total jerk. Ullrich's response was a joke, and anybody who
has looked at this group for a while would know that. Anyway, tommy,
David Ullrich knows more about math than you ever could in several of
your lifetimes. plz go read 55 math books!!
R.G. Vickson
Jake
Note quite what you were asking for, but the integrals int{0 to 1} (x^n/
n!) e^(1-x) dx = e - (1+1+1/2+1/3!+...+1/n!) seem to be of the same
spirit.
Zdislav V. Kovarik
I found it in the 29th Putnam Competition,
December 7, 1968, Problem A-1.
Somewhere in my old doodlings, I classified the integrals
A(m,n) = int(x^m*(1-x)^n/(1+x^2), 0, 1)
(m,n positive integers)
according to whether they belong to Q[pi] or Q[ln(2)], and with what
coefficients.
Cheers, ZVK(Slavek).
Gerry Myerson
"Zdislav V. Kovarik" <kov...@mcmaster.ca> wrote:
> Somewhere in my old doodlings, I classified the integrals
>
> A(m,n) = int(x^m*(1-x)^n/(1+x^2), 0, 1)
>
> (m,n positive integers)
>
> according to whether they belong to Q[pi] or Q[ln(2)], and with what
> coefficients.
See Backhouse, N. "Note 79.36. Pancake Functions and Approximations
to Pi." Math. Gaz. 79, 371-374, 1995.
--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
master1729
HA
Ullrich blundered !!
and this is one of his friend joining the
' cover-up operation '
what is so funny about doing an integral wrong ??
nothing !!
its just a blunder.
like when he arrived at 4 = 1.
getting personal to coverup a blunder is pathetic.
tommy1729
OwlHoot
You aren't half a muppet sometimes Amy/Tommy.
Ullrich has written books on analysis, for example:
http://www.amazon.com/Complex-Simple-Graduate-Studies-Mathematics/dp/0821844792
(by the looks of it, the kind of book you might benefit from reading)
Tonico
Well, this idiot Tommy/Ammy/whatever has elevated making an ass of
himself this to a rare form of art previously unknown in this NG.
An achievment, indeed.
Tonio
Rob Pratt
news:fb3281d0-a4ca-4651...@d34g2000vbm.googlegroups.com...
The folklore integral and extensions appear in the article "Approximations
to \pi Derived from Integrals with Nonnegative Integrands" by Stephen K.
Lucas in the February 2009 issue of The American Mathematical Monthly.
Rob Pratt
Dave L. Renfro
> The folklore integral and extensions appear in the article
> "Approximations to \pi Derived from Integrals with Nonnegative
> Integrands" by Stephen K. Lucas in the February 2009 issue
> of The American Mathematical Monthly.
Thanks. I didn't realize a paper on this had just been published.
I recently renewed/re-established my MAA membership (begun in
January 1974 and lapsed in 2002 or 2003 due to money problems),
but I've only gotten the most recent Amer. Math. Monthly and
the most recent College Math. J., and thus I haven't seen any
issues from the past couple of years. Typically, I wait until
the local university library binds the journal volumes before
I look at them, because I can't check them out unbound. Thus,
I knew about the 1995 Math. Gazette paper that Gerry Myerson
cited, but not the paper you cited.
Dave L. Renfro
Dave L. Renfro
> "It was hoped that some delightful integral such as
> the given one for 22/7 - pi would be found. Perhaps
> some clever reader will still discover an elegant
> integral for the desired 193/71 - e."
>
> Thus far I have only gotten up to the 1986 Pi Mu
> Epsilon Journal issues, so maybe at some later
> time something "delightful" was published. However,
> I thought I'd throw this out to the sci.math community
> and see what you people come up with.
To follow up on this, I've now gone through all the issues
up to Fall 2006 (last issue bound by the library at this time)
and I haven't seen any follow-up about this problem from
the Fall 1983 issue aside from what I've posted about that
appeared in the Fall 1984 issue.
Dave L. Renfro
klm
what are you talking about ?? 22/7 = pi ??
by cauchy ?
that is not funny and looks like bad math !
if a joke , a very very bad one and worthy of such a reply.
if intented as math even worse.
it doesnt matter what books are written by who !
perhaps that mathbook of ullrich contains more jokes ? it certainly doesnt look like a reference to me. if you want to joke about math you surely are not entitled to call the ones who are serious about math assholes or cranks.
lets turn it around;assume tommy said 22/7=pi, then you would reply that it proves that he is a crank. and when he doesnt he is a crank too ?!?
why ? just because you dont like this guy. i personally have no reasons to assume tommy doesnt know about integrals. and i guess neither have you. i know there is dispute about set theory but i also noticed tommy supports calculus and number theory. and that is not cranky or anything.
there is no reason to assume all enemies of zfc cant do integrals , especially if they appear as standard calculus supporters.
besides i dont know much about set theory but what i have read about tommys number theory in the 90's was very convincing and he has my respect for that. in fact i am tempted to believe him when he claims he has proof of the infinitude of prime twins. if anyone on sci.math can prove such a thing it must be him.
i know tommy can be sloppy sometimes or work with unfinished ideas or changing visions. he is sometimes vague or hard to follow , easily angered or impulsive. he is not a ' social mathematician ' , has problems working in groups (groups of people for clarity) and his work seems better when he is working alone.
but at the end tommy always gets a result.
always.
klm
master1729
typical.
no nice examples.
just a reference.
typical sci.math.
there probably isnt a nice example , just some messy expression in the reference above.
i guess this method of replying hides that as well.
keep up the mysterious attitude !
tommy1729