Re: Integrating over area of intersection
- To: mathgroup at smc.vnet.net
- Subject: [mg44142] Re: Integrating over area of intersection
- From: "Steve Luttrell" <luttrell at _removemefirst_westmal.demon.co.uk>
- Date: Fri, 24 Oct 2003 04:24:10 -0400 (EDT)
- References: <bn8esp$nle$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Here is an example of the sort of thing you can do.
First of all read in the Calculus`Integration` package which gives you the
Boole function for doing integrals over regions defined by inequalities:
<< Calculus`Integration`
Now integrate the function x^2*y^2 (as an example) over the region of
interest (as an example fJ = 3, fH = 2, f1 = 1/2, f2 = -4^(-1)}):
With[{fJ = 3, fH = 2, f1 = 1/2, f2 = -4^(-1)},
Integrate[Boole[x^2 + y^2 < fJ^2 &&
(x - f1)^2 + y^2 < fH^2 && (x - f2)^2 + y^2 <
fH^2]*x^2*y^2, {x, -Infinity, Infinity},
{y, -Infinity, Infinity}]]
which gives the result:
-((56359*Sqrt[247])/163840) + (11/3)*ArcCos[3/16] +
(35/6)*ArcSin[Sqrt[13/2]/4]
--
Steve Luttrell
West Malvern, UK
"Toni Danza" <nospam at yoohoo.com> wrote in message
news:bn8esp$nle$1 at smc.vnet.net...
> OK, I have three functions that are defined within their respective
circles.
> I would like to integrate over the intersection of the three circles.
>
> Here's what I have done:
> define region of integration:
> region = x^2 + y^2 < fJ^2 && (x - f1)^2 + y^2 < fH^2 && (x - f2)^2 +
y^2
> < fH^2
>
> Then I try to solve for the intersection using
>
> Reduce[region,{x,y}]
>
> and the result is something like (only works with numerical parameters...)
>
> -0.4<x<0.3 && sqrt(....)< y <sqrt(...) || -0.3<x<-0.2 && sqrt(....)< y
> <sqrt(...)
>
> How do I use this result to do integration over the region?
>