Re: Strange result with Integrate
- To: mathgroup at smc.vnet.net
- Subject: [mg38312] Re: Strange result with Integrate
- From: "David W. Cantrell" <DWCantrell at sigmaxi.org>
- Date: Thu, 12 Dec 2002 01:32:25 -0500 (EST)
- References: <at4bi0$elm$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Ray <rayfg at optonline.net> wrote:
> Can anyone explain the following results?
>
> In[8]:=
> r[t_]:=4-3Sin[t]
>
> In[9]:=
> \!\(NIntegrate[Sqrt[r[t]\^2 + \(r'\)[t]\^2], {t, 0, 2 \[Pi]}]\)
>
> Out[9]=
> 28.8142
This is correct.
> In[10]:=
> \!\(Integrate[Sqrt[r[t]\^2 + \(r'\)[t]\^2], {t, 0, 2 \[Pi]}] // N\)
>
> Out[10]=
> -18.9606
This is wrong. The exact (incorrect) result of this integration in my
version 4.1 is, after simplification,
2*(EllipticE[Pi/4, -48] + EllipticE[(3*Pi)/4, -48] -
25*(EllipticF[Pi/4, -48] + EllipticF[(3*Pi)/4, -48])).
The last two terms, involving incomplete elliptic integrals of the first
kind, are spurious. Omitting them, the answer would then be correct.
However, there is a simplification which I'm surprised that Mathematica
doesn't know:
2*(EllipticE[Pi/4, -48] + EllipticE[(3*Pi)/4, -48]) should simplify to
an expression with just a _single complete_ elliptic integral of the
second kind, namely
(#) 4*EllipticE[-48], a much nicer form of the exact (correct) answer.
But there's more of interest here!
Note that the radicand of the integrand simplifies to just 25 - 24*Sin[t].
So what do we get if we ask Mathematica to
Integrate[Sqrt[25 - 24*Sin[t]], {t, 0, 2*Pi}]? We find yet another error!
We get
2*EllipticE[Pi/4, -48] + 2*EllipticE[(3*Pi)/4, -48] +
4*EllipticF[I*ArcSinh[1/(4*Sqrt[3])], -48]
the last term of which is spurious (but, being imaginary, is different
from the spurious terms mentioned previously).
Next, let's see what we get if we ask for an indefinite integral:
Integrate[Sqrt[25 - 24*Sin[t]], t] yields -2*EllipticE[(Pi/2 - t)/2, -48].
Hooray! If we now use the Fundamental Theorem _ourselves_, we
do get a correct exact answer (finally!), albeit not in the simplified
form of (#). So I must now wonder why Mathematica got a spurious imaginary
term when it did the corresponding definite integral.
Finally, let's see what happens if we first transform the integral
ourselves using the standard substitution u = Sin[t]:
We get 2*Integrate[Sqrt[(25-24u)/(1-u^2)], {u, -1, 1}].
For this, Mathematica then gives
(28*I)*(EllipticE[1/49] - EllipticE[ArcSin[7], 1/49] +
EllipticF[ArcSin[7], 1/49] - EllipticK[1/49])
which, although messy (and looking as though it might be complex,
when it is in fact purely real), does happen to be correct! Hooray
again!
In summary, correct exact results can -- at least, sometimes -- be
obtained by assisting Mathematica ourselves, although these are not
as simple as (#). But we should not have to lead Mathematica by the
hand, so to speak. I will be very happy when such matters concerning
elliptic integrals are corrected.
David Cantrell
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