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Surface integral on a 3D region

  • To: mathgroup at smc.vnet.net
  • Subject: [mg111372] Surface integral on a 3D region
  • From: dr DanW <dmaxwarren at gmail.com>
  • Date: Thu, 29 Jul 2010 06:43:52 -0400 (EDT)

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[6]:=
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[7]:=
Chop[NIntegrate[Boole[region], {x, 0, 19.17}, {y, -3.5, 3.5},
   {z, -3.5, 3.5}]]

Out[7]= 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


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