       Re: Applying the Integration Function to a List Of Regions

• To: mathgroup at smc.vnet.net
• Subject: [mg88753] Re: Applying the Integration Function to a List Of Regions
• From: "David Park" <djmpark at comcast.net>
• Date: Fri, 16 May 2008 05:28:05 -0400 (EDT)
• References: <g0h4me\$lbl\$1@smc.vnet.net>

```regions = {{{x, 0, a}, {cx, 0, a + x}, {y, 0,
Sqrt[a^2 - cx^2 + 2 cx x - x^2]}, {cy, 0,
Sqrt[a^2 - cx^2 + 2 cx x - x^2] + y}}, {{x, 0, a}, {cx, 0,
a + x}, {y, Sqrt[a^2 - cx^2 + 2 cx x - x^2],
2 a - Sqrt[
a^2 - cx^2 + 2 cx x - x^2]}, {cy, -Sqrt[
a^2 - cx^2 + 2 cx x - x^2] + y,
Sqrt[a^2 - cx^2 + 2 cx x - x^2] + y}}, {{x, 0, a}, {cx, 0,
a + x}, {y, 2 a - Sqrt[a^2 - cx^2 + 2 cx x - x^2],
2 a}, {cy, -Sqrt[a^2 - cx^2 + 2 cx x - x^2] + y, 2 a}}, {{x, a,
2 a}, {cx, -a + x, 2 a}, {y, 0,
Sqrt[a^2 - cx^2 + 2 cx x - x^2]}, {cy, 0,
Sqrt[a^2 - cx^2 + 2 cx x - x^2] + y}}, {{x, a, 2 a}, {cx, -a + x,
2 a}, {y, Sqrt[a^2 - cx^2 + 2 cx x - x^2],
2 a - Sqrt[
a^2 - cx^2 + 2 cx x - x^2]}, {cy, -Sqrt[
a^2 - cx^2 + 2 cx x - x^2] + y,
Sqrt[a^2 - cx^2 + 2 cx x - x^2] + y}}, {{x, a, 2 a}, {cx, -a + x,
2 a}, {y, 2 a - Sqrt[a^2 - cx^2 + 2 cx x - x^2],
2 a}, {cy, -Sqrt[a^2 - cx^2 + 2 cx x - x^2] + y, 2 a}}}

Integrate[1, ##] & @@@ regions

--
David Park
djmpark at comcast.net
http://home.comcast.net/~djmpark/

"John Snyder" <jsnyder at wi.rr.com> wrote in message
news:g0h4me\$lbl\$1 at smc.vnet.net...
> Assume that I have already determined a list of 4 dimensional regions as
> follows:
>
> regions={{{x,0,a},{cx,0,a+x},{y,0,Sqrt[a^2-cx^2+2 cx
> x-x^2]},{cy,0,Sqrt[a^2-cx^2+2 cx
> x-x^2]+y}},{{x,0,a},{cx,0,a+x},{y,Sqrt[a^2-cx^2+2 cx x-x^2],2
> a-Sqrt[a^2-cx^2+2 cx x-x^2]},{cy,-Sqrt[a^2-cx^2+2 cx
> x-x^2]+y,Sqrt[a^2-cx^2+2 cx x-x^2]+y}},{{x,0,a},{cx,0,a+x},{y,2
> a-Sqrt[a^2-cx^2+2 cx x-x^2],2 a},{cy,-Sqrt[a^2-cx^2+2 cx x-x^2]+y,2
> a}},{{x,a,2 a},{cx,-a+x,2 a},{y,0,Sqrt[a^2-cx^2+2 cx
> x-x^2]},{cy,0,Sqrt[a^2-cx^2+2 cx x-x^2]+y}},{{x,a,2 a},{cx,-a+x,2
> a},{y,Sqrt[a^2-cx^2+2 cx x-x^2],2 a-Sqrt[a^2-cx^2+2 cx
> x-x^2]},{cy,-Sqrt[a^2-cx^2+2 cx x-x^2]+y,Sqrt[a^2-cx^2+2 cx
> x-x^2]+y}},{{x,a,2 a},{cx,-a+x,2 a},{y,2 a-Sqrt[a^2-cx^2+2 cx x-x^2],2
> a},{cy,-Sqrt[a^2-cx^2+2 cx x-x^2]+y,2 a}}};
>
> I want to integrate over each of these regions using an integrand of 1. I
> want my output to be as follows:
>
> {Integrate[1,{x,0,a},{cx,0,a+x},{y,0,Sqrt[a^2-cx^2+2 cx
> x-x^2]},{cy,0,Sqrt[a^2-cx^2+2 cx
> x-x^2]+y}],Integrate[1,{x,0,a},{cx,0,a+x},{y,Sqrt[a^2-cx^2+2 cx x-x^2],2
> a-Sqrt[a^2-cx^2+2 cx x-x^2]},{cy,-Sqrt[a^2-cx^2+2 cx
> x-x^2]+y,Sqrt[a^2-cx^2+2 cx x-x^2]+y}],Integrate[1,{x,0,a},{cx,0,a+x},{y,2
> a-Sqrt[a^2-cx^2+2 cx x-x^2],2 a},{cy,-Sqrt[a^2-cx^2+2 cx x-x^2]+y,2
> a}],Integrate[1,{x,a,2 a},{cx,-a+x,2 a},{y,0,Sqrt[a^2-cx^2+2 cx
> x-x^2]},{cy,0,Sqrt[a^2-cx^2+2 cx x-x^2]+y}],Integrate[1,{x,a,2
> a},{cx,-a+x,2
> a},{y,Sqrt[a^2-cx^2+2 cx x-x^2],2 a-Sqrt[a^2-cx^2+2 cx
> x-x^2]},{cy,-Sqrt[a^2-cx^2+2 cx x-x^2]+y,Sqrt[a^2-cx^2+2 cx
> x-x^2]+y}],Integrate[1,{x,a,2 a},{cx,-a+x,2 a},{y,2 a-Sqrt[a^2-cx^2+2 cx
> x-x^2],2 a},{cy,-Sqrt[a^2-cx^2+2 cx x-x^2]+y,2 a}]}
>
> How can I do that without having to set up each of the integrals manually?
> I am looking for some way to do something like:
>
> Integrate @@ regions
>
> or
>
> Integrate @@@ regions
>
> But I can't figure out how to incorporate the 1 as the integrand when I
> try
> to set this up automatically.
>
> There must be a way?
>
> Thanks,
>
> John
>
>

```

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