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/

**References**:**Re: piecewise integration***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>