Re: Bug in Integrate?

*To*: mathgroup at yoda.ncsa.uiuc.edu*Subject*: Re: Bug in Integrate?*From*: uunet!decide!bobk (Bob Korsan)*Date*: Wed, 15 Aug 90 10:32:05 PDT

You write ----- Begin Included Message ----- >From saxony!uunet!yoda.ncsa.uiuc.edu!mathgroup-adm Tue Aug 14 20:59:55 1990 From: uunet!mathnx.byu.edu!smithw (William V. Smith) To: mathgroup at yoda.ncsa.uiuc.edu Subject: Bug in Integrate? This problem came up in some instructional materials being developed here. If any of you know what's going on here let me know. I told the person who handed it to me, that it might be related to the rather awkward way Mathematica handles square roots. So, try doing this integration problem: In[1]:= a Integrate[Sqrt[(1-Cos[theta])^2+Sin[theta]^2],{theta,0,2 Pi}] Out[1]:= 0 (*this is incorrect*) In[2]:= a NIntegrate[Sqrt[(1-Cos[theta])^2+Sin[theta]^2],{theta,0,2 Pi}] Out[2]:= 8. a (*this is correct*) Date: Fri, 10 Aug 90 23:24:01 MDT From: smithw at mathnx.byu.edu (William V. Smith) To: mathgroup at yoda.ncsa.uiuc.edu Subject: Re: Another mathematica bug? Whoops! I just got a chance to look at this one again and it appears that NEITHER Integrate nor NIntegrate give the correct answer. I get 4aSqrt[2] working it out by hand. Sorry about that. Its been a busy day. Somebody want to check this? ----- End Included Message ----- Well, you are correct, the trouble is with Sqrt. If you look at the integrand and simplify In[1]:=Simplify[Sqrt[(1-Cos[t])^2+Sin[t]^2]/.Sin[t]^2->1-Cos[t]^2] Out[1]:=Sqrt[2 - 2 Cos[t]] and then In[2]:=Integrate[Sqrt[1-Cos[t]],t] Out[2]:= -2 Sqrt[2] ----------------------- 2 Sin[t] Sqrt[1 + -------------] 2 (1 + Cos[t]) If you plot this you will notice that since Mathematica always (EVEN WHEN IT SHOULDN'T) takes the positive square root, the result is always negative in the interval [0,2Pi]. Thus, the integrand evaluated at both end points is the same and subtracts to zero. This is a MAJOR BUG in version 1.2.1 and should be corrected immediately. Peace/Bob =============================================================================== | "The range of what we think and do | Bob Korsan | | is limited by what we fail to notice. | Decisions, Decisions! | | And because we fail to notice that we | 2136 Lyon Avenue | | fail to notice there is little we can | Belmont, CA 94002-1639 | | do to change until we notice how | 1(415) 595-1206 | | failing to notice shapes our thoughts | bobk at decide.UUCP | | and deeds." | korsan at pa.reuter.com | | -- R. D. Laing | {sun,uunet}!saxony!decide!bobk | ===============================================================================