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Integration over a Tetrahedon


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Calculate the integral of the function

$\displaystyle f(x,y,z)=x
$

over the tetrahedron

$\displaystyle T:\quad x,y,z \ge 0,\quad x+y
+\frac{z}{2}\le 1 \ .
$

The first step is to describe $ T$ as elementary region:

$\displaystyle T:\quad 0\le x\le 1,\quad 0\le y\le 1-x \,,\quad
0\le z\le 2\left(1-x-y\right)\,.
$

\includegraphics[width=0.45\moimagesize]{bsp_vol_tetraeder1}

Now the integral can easily be calculated by Fubini's theorem:

$\displaystyle \int\limits_T f$ $\displaystyle =$ $\displaystyle \int\limits_0^1\left(\int\limits_0^{1-x}
\left(\int\limits_0^{2(1-x-y)}x\,
dz\right) dy \right) dx$  
  $\displaystyle =$ $\displaystyle \int\limits_0^1\left(\int\limits_0^{1-x}2x\left(1-x-y\right)\,
dy\right)dx$  
  $\displaystyle =$ $\displaystyle \int\limits_0^1\left( x-2x^2+x^3 \right)\, dx
=\frac{1}{12} \,.$  

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  automatisch erstellt am 20.  8. 2008