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Integral of Triangular Surface

Date: 8/9/96 at 12:27:0
From: Anonymous
Subject: Integral of Triangular Surface

Is it possible to numerically integrate the following?  
Integral over S { 1/3 (x^3 i + y^3 j + 0 k) . n} dS where n is the 
unit normal to the surface which is available. The surface S is a 
triangle in a plane.

Date: 9/1/96 at 0:36:8
From: Doctor Jerry
Subject: Re: Integral of Triangular Surface

It looks like the integral you have written is a surface integral in 
standard form.  The vector n would be a constant since S is a 
triangle.  I interpret the 1/3 as a scalar multiple of the vector 
(x^3,y^3,0).  I think you don't need to do numerical integration.  It 
would be quite easy to give a parametrization of the triangle and then 
do an easy exact integration.

It appears, therefore, that your question is: how to parametrize the 
triangle S.  Suppose a, b, and c are vectors specifying the vertices 
of the triangle.  Then b-a and c-a are vectors from a and lying along 
the sides of the triangle.  You could take n equal to b-a cross c-a, 
afterwards dividing by the length of the cross product.

The parametrization of the triangle is 

r(u,v) = u(b-a) + v(c-a)

where (u,v) varies over the triangle in the (u,v)-plane bounded by
(0,0), (1,0), and (0,1).

Is this understandable?  Please write back if my comments aren't 

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
Associated Topics:
High School Calculus
High School Geometry
High School Triangles and Other Polygons

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