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From: "Tim Behrendsen" <tim@airshields.com>
Subject: Re: What's the best language to start with? [was: Re: Should I learn
 C or Pascal?]
Date: 1996/09/28
Message-ID: <01bbad7e$53e93ea0$32ee6fcf@timhome2>#1/1
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references: <01bb8df1$2e19d420$87ee6fce@timpent.airshields.com>
   <slrn52tn5u.8d.mdw@excessus.demon.co.uk>
 <322f864d.42836625@news.demon.co.uk>
 <01bb9bf9$61e9e0e0$87ee6fce@timpent.airshields.com>
 <50sj6q$aci@mtinsc01-mgt.ops.worldnet.att.net>
 <01bb9d25$9cb3cb00$32ee6fcf@timhome2>
 <50v6k3$soo@mtinsc01-mgt.ops.worldnet.att.net>
 <01bb9ded$cd0fdf00$32ee6fcf@timhome2> <5136on$7qj@goanna.cs.rmit.edu.au>
 <01bb9f26$36c870e0$87ee6fce@timpent.a-sis.com>
 <1996Sep24.133312.9745@ocsystems.com> <52gvu3$jhb@rumors.ucr.edu>
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newsgroups: comp.lang.c,comp.lang.c++,comp.unix.programmer,comp.lang.ada
Date: 1996-09-28T00:00:00+00:00
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Tom Payne <thp@cs.ucr.edu> wrote in article <52gvu3$jhb@rumors.ucr.edu>...
> In comp.lang.c++ Joel VanLaven <jvl@ocsystems.com> wrote:
> [...]
> :   Actually, not all functions are integrable.  The most complete and
> : irrefutable definition of the integral of a function to my knowledge is
:
> 
> : Given a partition P of [a,b] 
> :           (P is a finite subset of [a,b] including a and b)
> : P={x0,x1,x2,...xn} such that a=x0, b=xn, and x(j+1)>xj
> : call Mj the lub(f([x(j-1),xj]) (least upper bound)
> : call mj the glb(f([x(j-1),xj]) (greatest lower bound)
> 
> : The number Uf(P)=SUM(Mj(xj-x(j-1))) 1<=j<=n 
> : is called the P upper sum for f
> 
> : The number Lf(P)=SUM(mj(xj-x(j-1))) 1<=j<=n
> : is called the P lower sum for f
> 
> : The unique number I that satisfies the inequality
>   ^^^^^^^^^^^^^^^^^^^
>    if it exists
> 
> : Lf(P)<=I<=Uf(P) for all possible P of [a,b]
> : is called the definite integral of f from a to b.
> 
> Actually, there is a significant generalization, called Lebesgue
> integration, that can be found in any graduate text on Real Analysis
> (e.g., those by Royden and by Rudin).  Nevertheless, if you believe in
> the axiom of choice, there are still functions that are not integrable
> (but they get pretty weird).

Somewhat off the subject (but way off the subject of comp.lang.*), the
site http://www.integrals.com (sponsored by Mathematica?) that allows
you to type an integral in and it will try and evaluate it.  If it
can't do it, it's either not integrable or their algorithm doesn't
support it (they ask you to e-mail them if you think it can be done).
I managed to break it right off by

   integral X^X dx

Which I *think* is integrable (at least I remember figuring it out
in college, but that was a long time ago and I may have done it
wrong. :-) )

-- Tim Behrendsen (tim@a-sis.com)