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RE: Integration of very simple partwise defined functions fails
*To*: mathgroup at smc.vnet.net
*Subject*: [mg15775] RE: [mg15720] Integration of very simple partwise defined functions fails
*From*: "Ersek, Ted R" <ErsekTR at navair.navy.mil>
*Date*: Sun, 7 Feb 1999 02:03:53 -0500 (EST)
*Sender*: owner-wri-mathgroup at wolfram.com
Daniel Oberfeld wrote:
___________________
I was quite surprised that Mathematica will not evaluate any Integrals
of simple functions like
f[x_/;x<=2]:=1
f[x_/;x>2]:=2
Integrate[f[x],{x,0,1}]
NIntegrate also complains about singularities.
Do I miss something? There must be a fast way out...
________________
I don't know what you can do to get Integrate to deal with this. But
NIntegrate[f,{x,x0,x1,x2,...xi,xj}]effectively breaks up the integral
into
pieces along [x0,x1], [x1,x2], ...,[x,xi]. So include the point of
discontinuity in the range of integration.
In[8]:=
NIntegrate[f[x],{x,0,2,5}]
Out[8]=
8.
If you don't know where the discontinuities are you can write some code
that does it by using NDSolve on the equivalent Diff. Eq.
In[9]:=
NIntegrate2[f_,{x_,a_,b_}]:=
Module[{y},
y[b]/.NDSolve[{y'[x]==f,y[a]==0},y,{x,a,b}][[1]]
]
In[10]:=
NIntegrate2[f[x],{x,0,5}]
Out[10]=
8.00001
Regards,
Ted Ersek
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