Mathematik-Online lexicon: Integration over a Tetrahedon

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	Mathematik-Online lexicon:
	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

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