Mo logo [home] [lexicon] [problems] [tests] [courses] [auxiliaries] [notes] [staff] german flag

Mathematik-Online lexicon:

Integration over a Tetrahedon

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z overview

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)\,.


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

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



  automatisch erstellt am 20.  8. 2008