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Re: Simple integral wrong
*To*: mathgroup at smc.vnet.net
*Subject*: [mg25073] Re: Simple integral wrong
*From*: Paul Cally <cally at kronos.maths.monash.edu.au>
*Date*: Thu, 7 Sep 2000 22:28:17 -0400 (EDT)
*Organization*: Monash University
*References*: <8opmr5$15@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
Paul Cally wrote:
> Try integrating | cos u| from u=0 to u = Pi x. Despite the integrand
> being everywhere
> non-negative, Mathematica 4 gives a result which jumps DOWNWARDS by 2 at
>
> x=1/2, 3/2, 5/2, .... I thought these simple integration errors had been
> sorted out by
> Wolfram years ago!
>
> Paul Cally
>
> --
>
> +--------------------------------------------------------------------------+
> |Assoc Prof Paul Cally | Ph: +61 3 9905-4471 |
> |Dept of Mathematics & Statistics | Fax: +61 3 9905-3867 |
> |Monash University | paul.cally at sci.monash.edu.au |
> |PO Box 28M, Victoria 3800 | |
> |AUSTRALIA | http://www.maths.monash.edu.au/~cally/ |
> +--------------------------------------------------------------------------+
I have had a number or queries about this. Sorry I hadn't made it clearer.
Here is
the full story:
A couple of people told me that
Plot[Integrate[Abs[Cos[u]], {u, 0, x Pi]}], {x, 0, 3}]
works fine. The result is monotonic increasing as expected.
But try
Plot[Evaluate[Integrate[Abs[Cos[u]],{u,0,Pi*x}]],{x,0,3}]
and see what happens! The evaluate forces Mathematica to do the
integral symbolically. It was doing it numerically without the Evaluate.
Or just type
Integrate[Abs[Cos[u]],{u,0,Pi x}]
Mathematica 4 returns
2
Out[1]= Sqrt[Cos[Pi x] ] Tan[Pi x]
(Actually, I don't think Mathematica 3 can do it at all.) This
plots as a saw-tooth. The true solution should be
Sqrt[Cos[Pi x]^2] Tan[Pi x] + 2 Floor[x + 1/2]
Mathematica misses the step functions necessary to make the solution
continuous.
Thanks for your interest,
Paul Cally
--
+--------------------------------------------------------------------------+
|Assoc Prof Paul Cally | Ph: +61 3 9905-4471 |
|Dept of Mathematics & Statistics | Fax: +61 3 9905-3867 |
|Monash University | paul.cally at sci.monash.edu.au |
|PO Box 28M, Victoria 3800 | |
|AUSTRALIA | http://www.maths.monash.edu.au/~cally/ |
+--------------------------------------------------------------------------+
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