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RE: Re: How to integrate over a constrained domain

• To: mathgroup at smc.vnet.net
• Subject: [mg34246] RE: [mg34217] Re: [mg34203] How to integrate over a constrained domain
• From: "DrBob" <majort at cox-internet.com>
• Date: Sat, 11 May 2002 04:05:09 -0400 (EDT)
• Reply-to: <drbob at bigfoot.com>
• Sender: owner-wri-mathgroup at wolfram.com

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: [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

• Follow-Ups:
  • Re: RE: Re: How to integrate over a constrained domain
    • From: Murray Eisenberg <murraye@attbi.com>