Re: Fw: Integrate...

*To*: mathgroup at smc.vnet.net*Subject*: [mg44370] Re: Fw: Integrate...*From*: "David W. Cantrell" <DWCantrell at sigmaxi.org>*Date*: Fri, 7 Nov 2003 05:16:13 -0500 (EST)*References*: <bob5ac$qd3$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

"Christos Argyropoulos M.D." <argchris at otenet.gr> wrote: > I think the problem goes deeper than simply mixing symbolic/numerical > stuff. You're absolutely right. Of course, what Jens-Peer said, seconded by Curt, is good advice. But it in no way explains the huge discrepancy between the incorrect -2.82839 and the correct 0.000037373 . > Try symbolic integration of the original function [snip] > Results of the original, simplified versions and the numerical > integration agree for 0<=a<=2.35 and 0<=b<2.35 > After they diverge dramatically. I suspect this is a bug somewhere > inside Integrate, but I do not have the guts to evaluate the > integral by hand to see, if the expression returned for symbolic a,b > are correct. I guess that depends on your precise definition of "bug". Here's what's going on: Integrate[Abs[Cos[r] + Sin[r]], {r, a, b}] gives a messy expression. Then Assuming[a >= 0 && b > a, FullSimplify[%]] yields ((-a + b)*(((-Cos[a] + Sin[a])*Sqrt[1 + Sin[2*a]])/ (Cos[a] + Sin[a]) + ((Cos[b] - Sin[b])* Sqrt[1 + Sin[2*b]])/(Cos[b] + Sin[b])))/(a - b) (* Aside: Why on earth, since I had specified that b > a, doesn't FullSimplify cancel (-a + b) with (a - b) to give just -1 ? *) Next % /. {a -> 235/100, b -> 236/100} gives a slightly messy, exact expression, after which N[%] yields the absurd result of -2.82839 . The problem lies in the fact that the result given for the definite integral with symbolic limits is incorrect (in the same way that I discussed in my first response to Selwyn in the recent thread "Integrate 5.0"). Namely, although the integrand is _continuous_, Mathematica's antiderivative is not, having spurious discontinuities. One of those lies at 3*Pi/4, which is between 2.35 and 2.36. It is the source of the absurd result, which is then, not accidentally, very close to -2*Sqrt[2]. An antiderivative for Abs[Cos[x]+Sin[x]] which is continuous for all real x, obtained by hand, is Sqrt[2](2 Floor[x/Pi + 1/4] - (-1)^Floor[x/Pi + 1/4]Cos[x + Pi/4]) Of course, having that, one can easily write an expression for the definite integral with symbolic limits which is valid for _all_ real a and b. It's too bad that Mathematica can't do that itself. David Cantrell > ----- Original Message ----- > From: "Sampo Smolander" <sampo.smolander+news at helsinki.fi> To: mathgroup at smc.vnet.net > Subject: [mg44370] Integrate... > > > Does anybody know why > > > > Integrate[Abs[1.0*Cos[r] + 1.0*Sin[r]], {r, 2.35, 2.36}] > > > > gives > > > > -2.82839 > > > > whereas without those 1.0's, > > > > Integrate[Abs[Cos[r] + Sin[r]], {r, 2.35, 2.36}] > > > > gives the correct > > > > 0.000037373