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MathGroup Archive 2006

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Re: Re: piecewise integration

  • To: mathgroup at smc.vnet.net
  • Subject: [mg67052] Re: [mg66999] Re: piecewise integration
  • From: "Chris Chiasson" <chris at chiasson.name>
  • Date: Thu, 8 Jun 2006 04:53:28 -0400 (EDT)
  • References: <20060605102611.774$dR_-_@newsreader.com> <200606061028.GAA20748@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

> I confess that I never noticed that Integrate and DiracDelta in
> Mathematica behaved like this at end points.  It seems to me that the
> Piecewise approach, which assumes that boundary points are treated
> the same as interior points, is the more natural. But Chris obviously
> was not interested in the answer to this particular problem but in
> more general matters. It is trivial to modify the behaviour of the
> package in this respect (by adding ones own rules for handling
> DiracDelta) to make it conform with what Mathematica does, if one
> really wanted to.  But my main point was that the package is
> interesting in its own right and it seems to me that anyone seriously
> interested in this topic would have already taken a look at it. Why,
> even people not seriously interested in it, like myself, have done so
> and found interesting and instructive things in it.
>
> Andrzej

I think Mathematica's Integrate does this to preserve the identity:
Integrate[f[x],{x,a,c}]==Integrate[f[x],{x,a,b}]+Integrate[f[x],{x,b,c}]

There is a danger of violating this when using PiecewiseIntegrate:

In[1]:=
<<PiecewiseIntegrate.m
load[x_]=-9*10^3*DiracDelta[x]-
    Piecewise[{{x*10*(10^3/3),0\[LessEqual]x\[LessEqual]3}}]-6*10^3*
      DiracDelta[x-5]
Integrate[load[x],{x,-10,10}]
PiecewiseIntegrate[load[x],{x,-10,10}]
{PiecewiseIntegrate[load[x],{x,-10,0}],PiecewiseIntegrate[load[x],{x,0,5}],
  PiecewiseIntegrate[load[x],{x,5,10}]}
Plus@@%
Out[2]=
-6000*DiracDelta[-5 + x] - 9000*DiracDelta[x] -
Piecewise[{{(10000*x)/3, 0 <= x <= 3}}]
Out[3]=
-30000
Out[4]=
-30000
Out[5]=
{-9000,-30000,-6000}
Out[6]=
-45000
-- 
http://chris.chiasson.name/


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