I have a common question concerning the three integral questions below.
I don't know how to deal with terms like $|x|$. I can parameterize both the region of integration or the integrand, but do I have to consider the two cases for $x < 0$ or $x \geq 0$
One other thing, isn't the region $|x|+|y|+|z| \leq 1$ suppose to be called an octahedron. So when i integrate over this region, should it not be two times the integral over a tetrahedron.
Thank you in advance.
Integrate the function $f(x,y,z)={x}^{2}+{y}^{2}+{z}^{2}$ over the tetrahedron
$\ E=\{(x,y,z): |x|+|y|+|z| \leq 1\} $
Integrate the function $f(x,y,z)=|z|$ over the tetrahedron
$\ E=\{(x,y,z): |x|+|y|+|z| \leq 1\} $
Integrate $f(x,y,z)=|z|$ over the surface of the sphere
$\ E=\{(x,y,z):{x}^{2}+{y}^{2}+{z}^{2}=1\} $