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[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 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[3]= 7/256 (4288 + 35 Sqrt[401]) Pi Maxim Rytin m.r at inbox.ru