[math-fun] Re help with definite integral
The other day I asked for help in evaluating a definite integral Let h = 1/2, to save space. Int_{x = 0..h} Int_{r=-h-x..h-x} Int_{y=0..h} Int_ {s=-h-y..h-y} sqrt(r^2+s^2) dx dr dy ds = ? Thanks to David Cantrell, Eric Weisstein, Steve Finch, Emeric Deutsch and others who replied. To summarize the results: Integrals like this can be found in the books Finch, Encyclop. of Math. Constants (CUP); Santalo, Integral Geometry (Addison-Wesley) There is a web page in Spanish: <http://groups.google.com/group/es.ciencia.matematicas/browse_frm/thread/efec... d222ada58ab7> Mathematica can do this integral, but you have to put the variables in the right order, and you need a machine with the right operating system (it worked on a Sun Linux 64-bit machine, but - with the same version of Mma - failed on an Intel-based Linux 64-bit machine) Best regards Neil
A similar problem appeared in the Technology Review Puzzle Corner, circa 1975: Consider a domino, two adjacent 1x1 squares. What's the average distance between a random point in the left square, and a random point in the right square? We did it by hand, and the answer was in closed form, but a mess. I found that Macsyma of that era could do most of the integral, with hand-holding, but ran into a problem at the final level. It turns out that being able to answer the puzzle for two random points in a rectangle is enough to answer it for any two (orthogonally oriented) rectangles in the plane. Math needs to deal better with made-up functions like this. The closed form answers are something of a distraction: too complex, not especially informative. Consider the 3-D version of this puzzle, which might be relevant in modeling random gasses. Rich ________________________________________ From: math-fun-bounces@mailman.xmission.com [math-fun-bounces@mailman.xmission.com] On Behalf Of N. J. A. Sloane [njas@research.att.com] Sent: Thursday, February 28, 2008 9:40 AM To: math-fun@mailman.xmission.com Cc: njas@research.att.com Subject: [math-fun] Re help with definite integral The other day I asked for help in evaluating a definite integral Let h = 1/2, to save space. Int_{x = 0..h} Int_{r=-h-x..h-x} Int_{y=0..h} Int_ {s=-h-y..h-y} sqrt(r^2+s^2) dx dr dy ds = ? Thanks to David Cantrell, Eric Weisstein, Steve Finch, Emeric Deutsch and others who replied. To summarize the results: Integrals like this can be found in the books Finch, Encyclop. of Math. Constants (CUP); Santalo, Integral Geometry (Addison-Wesley) There is a web page in Spanish: <http://groups.google.com/group/es.ciencia.matematicas/browse_frm/thread/efec... d222ada58ab7> Mathematica can do this integral, but you have to put the variables in the right order, and you need a machine with the right operating system (it worked on a Sun Linux 64-bit machine, but - with the same version of Mma - failed on an Intel-based Linux 64-bit machine) Best regards Neil _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Tue, Mar 4, 2008 at 4:26 PM, Schroeppel, Richard <rschroe@sandia.gov> wrote:
A similar problem appeared in the Technology Review Puzzle Corner, circa 1975:
Consider a domino, two adjacent 1x1 squares. What's the average distance between a random point in the left square, and a random point in the right square?
We did it by hand, and the answer was in closed form, but a mess. I found that Macsyma of that era could do most of the integral, with hand-holding, but ran into a problem at the final level.
Just for the record, Mma (6.0) does fine: d[x1_,y1_,x2_,y2_] := Sqrt[ (x1-x2)^2+(y1-y2)^2 ] Integrate[ d[x1,y1,x2,y2], {x1,-1,0},{x2,0,1},{y1,0,1},{y2,0,1} ] (116 - 8 Sqrt[2] - 20 Sqrt[5] + 140 ArcCsch[2] - 40 ArcSinh[1] + 80 ArcSinh[2] + Log[32] + 10 Log[-1 + Sqrt[5]] - 15 Log[123 + 55 Sqrt[5]]) / 120 --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.
participants (3)
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Michael Kleber -
N. J. A. Sloane -
Schroeppel, Richard