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MathGroup Archive 2007

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Re: Re: Integrate question

  • To: mathgroup at smc.vnet.net
  • Subject: [mg77749] Re: [mg77685] Re: Integrate question
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Sat, 16 Jun 2007 03:21:47 -0400 (EDT)
  • References: <f4omvd$7qh$1@smc.vnet.net><f4r10i$639$1@smc.vnet.net> <200706150825.EAA15568@smc.vnet.net>

I think Jens simply got confused: it is the derivative f' that needs  
to be assumed to be continuous, not the function f. What "continuous"  
in the "other CAS" presumably means is "assume that the integrand is  
continuous", and in this case it is the derivative of f that is the  
integrand.

However, I have to confess, I fail to see any serious use for this  
(in the case of an undefined fucntion f). Does not having  
"continuous" in the integral amount to anything  more than simply the  
instruction: do not evaluate this integral using the Fundamental  
Theorem of Calculus? Is there any other practical aspect to it? In  
Mathematica you also have a number of ways of preventing evaluation  
if you don't want it and I don't see any obvious advantage in the way  
the "other CAS" does this, except perhaps that of giving us the  
illusion of greater mathematical rigour. But then perhaps I am  
failing to see something important?

Andrzej Kozlowski


On 15 Jun 2007, at 17:25, dimitris wrote:

> I agree with this!
> I should have posted my question differently.
>
> In[2]:=
> Integrate[D[f[x], x], {x, a, b}]
>
> Out[2]=
> -f[a] + f[b]
>
>
>> int(D(y)(x),x=a..b);
>
>                               b
>                              /
>                             |
>                             |   D(y)(x) dx
>                             |
>                            /
>                              a
>
>> int(D(y)(x),x=a..b,'continuous');
>
>                              -y(a) + y(b)
>
> Integrate[f'[x],{x,a,b}] is NOT always f[b]-f[a].
>
> In[30]:= g[x_] = Integrate[ArcTan[Tan[x/2]], x];
>
> In[34]:= Integrate[Derivative[1][g][x], {x, 0, 3*(Pi/2)}]
> Out[34]= Pi^2/16
>
> In[37]:= FullSimplify[(D[g[x], x] /. x -> 3*(Pi/12)) - (D[g[x], x] /.
> x -> 0)]
> Out[37]= Pi/8
>
> Dimitris
>
>  /  Jens-Peer Kuska       :
>> Hi dimitris,
>>
>> this is a very valuable question:
>>
>>> I am neither software enginner, nor pure mathematician but this
>>> fundmental difference impressed me a lot! I am familiar
>>> with the issue of generic complex values in mathematica
>>> but here Mathematica "assumes" that typing Integrate[f'[x],{x,a,b}]
>>> f[x] is continuous in [a,b]?
>>
>>
>> and what happens with f'[x] if f[x] is *not* continuous in [a,b] ??
>> it does not exist on every point in[a,b]?
>> You can't write f'[x] if it does not exist
>> and writing it mean that f'[x] exist and that the integral will be
>> continuous.
>>
>> Regards
>>    Jens
>
>



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