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Re: RE: Re: How to integrate over a constrained domain

  • To: mathgroup at smc.vnet.net
  • Subject: [mg34256] Re: [mg34246] RE: [mg34217] Re: [mg34203] How to integrate over a constrained domain
  • From: Murray Eisenberg <murraye at attbi.com>
  • Date: Sun, 12 May 2002 03:25:59 -0400 (EDT)
  • Organization: Mathematics & Statistics, Univ. of Mass./Amherst
  • References: <200205110805.EAA21912@smc.vnet.net>
  • Reply-to: murray at math.umass.edu
  • Sender: owner-wri-mathgroup at wolfram.com

It's NOT undocumented -- in the Help Browser, go to Add-ons, Standard
Packages, Calculus, Integration.

However, it IS missing from the Master Index!

DrBob wrote:
> 
> Boole --- another undocumented feature.   Sigh...
> 
> Bobby
> 
> -----Original Message-----
> From: BobHanlon at aol.com [mailto:BobHanlon at aol.com]
To: mathgroup at smc.vnet.net
> Subject: [mg34256] [mg34246] [mg34217] Re: [mg34203] How to integrate over a constrained
> domain
> 
> In a message dated 5/9/02 6:42:13 AM, maciej at maciejsobczak.com writes:
> 
> >Let's say I have a set on a (x,y) plane given by:
> >
> >x^2 + y^2 < r^2
> >
> >and I want to compute its area.
> >Yes, I know its Pi*r^2, but I want Mathematica tell me.
> >
> >As a generalization, I want to integrate over a domain given by one or
> >more
> >inequalities.
> >The problem above can be solved like this:
> >
> >Integrate[1, {x, -r, r}, {y, -Sqrt[r^2-x^2], Sqrt[r^2-x^2]}]
> >Simplify[%, {r>0}]
> >
> >which gives
> >
> >Pi r^2
> >
> >That's nice, but requires solving the inequality for y, which is not
> always
> >viable.
> >
> >It would be nice to have syntax like:
> >
> >Integrate[1, {x, y}, {x^2 + y^2 < r^2}]
> >
> >but it does not work (of course).
> >
> >How can I achieve what I want?
> 
> For specific numeric values it is easy
> 
> Needs["Calculus`Integration`"];
> 
> Table[{r,
> 
>     Integrate[Boole[ x^2+y^2<r^2] ,
> 
>       {x,-r,r}, {y,-r,r}]},
> 
>   {r,0,5}]
> 
> {{0, 0}, {1, Pi}, {2, 4*Pi}, {3, 9*Pi}, {4, 16*Pi},
> 
>   {5, 25*Pi}}
> 
> Bob Hanlon
> Chantilly, VA  USA

-- 
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.       
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street
Amherst, MA 01375


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