Re: Simplified Integration Problem

*To*: mathgroup at smc.vnet.net*Subject*: [mg35591] Re: Simplified Integration Problem*From*: "Carl K. Woll" <carlw at u.washington.edu>*Date*: Sun, 21 Jul 2002 01:01:10 -0400 (EDT)*Organization*: University of Washington*Sender*: owner-wri-mathgroup at wolfram.com

Arnold, The explanation for the difference in results when using Integrate versus NIntegrate lies in how the answers are computed. The function NIntegrate typically samples the integrand at a sufficient number of points to approximate the answer, using a strategy similar to but more sophisticated than the trapezoidal approximation. In general, NIntegrate will always get the correct answer, although it is possible that the answer produced by NIntegrate may not be very precise. The function Integrate, on the other hand, basically does what you or I would do. It tries to find the antiderivative of the integrand, and then, if there are limits involved, those limits are plugged into the antiderivative. Usually, this procedure works perfectly well, and the correct answer is produced. However, consider the case where the antiderivative has multiple branches. Then, Integrate must use the same branch when plugging in the limits into the integrand in order to produce the correct answer. Sometimes, Integrate will choose one branch for the upper limit, and a different branch for the lower limit. When this happens, Integrate will get the wrong answer. The message you should get from the above discussion is that whenever you want to compute a DEFINITE integral using Integrate, you should be aware that the resulting answer may be incorrect if the antiderivative of the integrand has multiple branches. On the other hand, for definite integrals, NIntegrate should always produce the correct answer. In other words, any time you are trying to compute a definite integral with Integrate, and the integrand looks complex, you probably should verify the answer using NIntegrate. Carl Woll Physics Dept U of Washington "Arnold Gregory Civ AFRL/SNAT" <Gregory.Arnold at wpafb.af.mil> wrote in message news:ah8p7q$avj$1 at smc.vnet.net... > Here's a slightly simplified version of the integration problem I presented the other day: > > int=Integrate[UnitStep[1 - t^2 - c^2*(-4*t + 8*t^3)^2], {t, -1, 1}]; > int /. c->0 Result:2 > int /. c->0. Result:0. > int /. c->3 Result:2 > int /. c->3. Result:0.28817 > > In fact, "int /. c-> any rational" always yields 2!!! > > NIntegrate gets the right answers: > NIntegrate[UnitStep[1 - t^2 - 0^2*(-4*t + 8*t^3)^2], {t, -1, 1}, > MinRecursion->5,MaxRecursion->30] Result: 2 > > NIntegrate[UnitStep[1 - t^2 - 2^2*(-4*t + 8*t^3)^2], {t, -1, 1}, > MinRecursion->5,MaxRecursion->30] Result: 0.28817 > > Plot[int,{c,-5,5}] also gets the right results. > > Any help or insight into why Integrate isn't working would be greatly appreciated! > > Greg > > >