Re: Correction on confusion with triple integral...

*To*: mathgroup at smc.vnet.net*Subject*: [mg32508] Re: [mg32488] Correction on confusion with triple integral...*From*: David Withoff <withoff at wolfram.com>*Date*: Thu, 24 Jan 2002 05:21:02 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

> My apologies. After reading Bob's message I realized I had a typo. The > limits on r were supposed to be [1/Cos[x],4], not [0,1/Cos[x]]. So, here is > the corrected problem... > > Please consider the Integral[r^2Sin[x]drdxdy] with limits as follows: > > [0,2Pi]for y, [0,ArcCos[1/4]] for x and [1/Cos[x],4] for r. Mathematica > returns 27Pi, which is correct. But, if instead of using ArcCos[1/4], I use > either ArcCos[0.25] or ArcCos[1/4]//N in the limits, Mathematica returns > 43Pi. > Any idea why? Multiple integrals are done as repeated single integrals, so Integrate[r^2*Sin[x], {x, 0, ArcCos[1/4]}, {y, 0, 2*Pi}, {r, 1/Cos[x], 4}] comes down to Integrate[2*Pi*(64/3 - Sec[x]^3/3)*Sin[x], {x, 0, ArcCos[1/4]}] which is done by taking limits of the corresponding indefinite integral: In[1]:= Integrate[2*Pi*(64/3 - Sec[x]^3/3)*Sin[x], x] Out[1]= 2*Pi*(-((Cos[x]*(64/3 - Sec[x]^3/3))/ (2*(-1 + 48*Cos[x] + 16*Cos[3*x]))) - (64*Cos[x]^4*(64/3 - Sec[x]^3/3))/(-1 + 48*Cos[x] + 16*Cos[3*x])) which has a removeable singularity at ArcCos[1/4]. Handling of singularities is difficult unless the exact location of the singularity is known, so there is a numerical problem if the location of the singularity is given by an inexact number like ArcCos[0.25]. The Integrate function should nevertheless be able to deal with this. I'm not intending to suggest that this behavior is intentional or inevitable. The fact that Integrate gives the wrong answer here should just be fixed. I presume that what you were asking, though, was why there might be any sort of difficulty here, and that's why -- it's numerical error related to inexact specification of the location of a singularity. Dave Withoff Wolfram Research