Re: 22/7 - pi integral and a sci.math challenge

"Dave L. Renfro" <renfr1dl@xxxxxxxxx> wrote in message
news:fb3281d0-a4ca-4651-80fe-0fdbcb762d9c@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
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.

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


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


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