       Re: Surface integral on a 3D region

• To: mathgroup at smc.vnet.net
• Subject: [mg111397] Re: Surface integral on a 3D region
• From: Maxim <m.r at inbox.ru>
• Date: Sat, 31 Jul 2010 02:38:57 -0400 (EDT)
• References: <i2rm1a\$r57\$1@smc.vnet.net>

```On Jul 29, 5:43 am, dr DanW <dmaxwar... at gmail.com> wrote:
> I have been working on a problem that involves integrating odd shapes,
> that is, shapes defined by surfaces that are not on constant
> coordinate planes.  The concept of working with regions defined by
> inequalities is new to me.  My shape is a cylinder with one normal and
> one oblique termination:
>
> In:=
> region = y^2 + z^2 <= 3.5^2 && 0 <= x && 0.64*(-15 + x) + 0.77*z <=
= 0;
>
> I was gratified to find that NIntegrate and Boole lets me do a volume
> integration:
>
> In:=
> Chop[NIntegrate[Boole[region], {x, 0, 19.17}, {y, -3.5, 3.5},
>    {z, -3.5, 3.5}]]
>
> Out= 577.267
>
> However, now I am faced with needing a surface integration.  Is there
> a Mathematica technique that I have not found to do this directly?  Of
> course, I am aware that for this problem that I can grind through the
> details of setting up nested integrations with variable limits of
> integration, but I am lazy and want Mathematica to do the work.
> Besides, if there is a general methodology, that would be far more
> valuable to me than the solution to one particular problem.
>
> Thanks for the help.
>
> Daniel

It's possible to rewrite a surface integral in terms of DiracDelta.
You can find a couple of examples here:
http://library.wolfram.com/infocenter/MathSource/5117/ .

region = {y^2 + z^2 <= (35/10)^2, 0 <= x,
64/100 (-15 + x) + 77/100 z <= 0};

Fold[PiecewiseIntegrate,
DiracDelta[#/Norm@ D[#, {{x, y, z}}]]&[Subtract @@ region[[#]]]*
Boole[And @@ Delete[region, #]],
{#, -Infinity, Infinity}& /@ {x, y, z}]& /@
Range@ Length@ region // Total // Simplify

Out= 7/256 (4288 + 35 Sqrt) Pi

Maxim Rytin
m.r at inbox.ru

```

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