Re: How to integrate over a constrained domain
- To: mathgroup at smc.vnet.net
- Subject: [mg34217] Re: [mg34203] How to integrate over a constrained domain
- From: BobHanlon at aol.com
- Date: Fri, 10 May 2002 03:04:59 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
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