Re: once more integrate
- To: mathgroup at smc.vnet.net
- Subject: [mg20793] Re: [mg20735] once more integrate
- From: Wolfgang Schadow <schadow at netcom.ca>
- Date: Sun, 14 Nov 1999 18:13:46 -0500 (EST)
- References: <199911100517.AAA19687@smc.vnet.net.>
- Sender: owner-wri-mathgroup at wolfram.com
Reply from Wolfram:
> In[4]:= Integrate[1/x^2,{x,-2,2}]
>
> Out[4]= Infinity (correct)
>
> In[5]:= Integrate[1/x^2,{x,-Abs[a],2}] /. a -> 2
>
> Out[5]= -1 (not correct)
>
> In[6]:= Integrate[1/x^2,{x,-Abs[a],2}]
>
> 1 1
> Out[6]= -(-) - ------
> 2 Abs[a]
>
> In[7]:= Integrate[1/x^2,{x,-a,a}]
>
> -2
> Integrate::idiv: Integral of x does not converge on {-a, a}.
>
> -2
> Out[7]= Integrate[x , {x, -a, a}]
>
> In the first case Mathematica gives the corrcet answer, in the
> second case it overlooks the singularity. In the third case the
> answer is at least not wrong.
>
> Wolfgang Schadow
>
> ========================================================================
> Wolfgang Schadow Phone: +1-604-222-1047 ext. 6453 (office)
> TRIUMF +1-604-875-6066 (home)
> Theory Group FAX: +1-604-222-1074
> 4004 Wesbrook Mall
> Vancouver, B.C. V6T 2A3 email: schadow at triumf.ca
> Canada www : http://www.triumf.ca/people/schadow
>
> ========================================================================
>
> ----------------------------------------------------------------------------
>
> Hello,
>
> Thank you for the email.
>
> 1) You cannot do a replacement after and integral as you have done. The
> reaon is that Mathematica evaluates the integral first, and then does the
> replacement.
>
> 2) If you make the integral in this case a function of a as follows:
>
> In[1]:= f[a_]:= Integrate[1/x^2, {x, -Abs[a], 2}]
>
> Then you can then evaluate the integral correctly by passing 2 to the
> integral before it is evaluated:
>
> In[2]:= f[2]
>
> Out[2]= Infinity
>
> In the following case:
>
> In[3]:= Integrate[1/x^2, {x, -Abs[a], 2}]
>
> You do get an incorrect answer, but it is the same answer as the one you
> would get if you did it by hand. The Fundamental Theorem of calculus is
> "dumb" in that it doesn't know there is a singularity at x=0. You can
> only use the Fundamental Theorem of calculus if you know that your
> function is well behaved and has no singularities.
>
> If you don't give an exact value for a bound, Mathematica will try to
> use the Fundamental Theorem of calculus, which gives a bad result in this
> case.
>
> As long as you specify an exact value for a before hand or in place of
> a, you will get the correct answer.
>
> Sincerely,
>
> Jeffrey Bryant
> Technical Support
> Wolfram Research, Inc.
> support at wolfram.com
========================================================================
Wolfgang Schadow Phone: +1-604-222-1047 ext. 6453 (office)
TRIUMF +1-604-875-6066 (home)
Theory Group FAX: +1-604-222-1074
4004 Wesbrook Mall
Vancouver, B.C. V6T 2A3 email: schadow at triumf.ca
Canada www : http://www.triumf.ca/people/schadow
========================================================================
- References:
- once more integrate
- From: Wolfgang Schadow <schadow@netcom.ca>
- once more integrate