I need to calculate $ \int_B f(x)dx $ for: $f(x,y,z)=x*y*z$
The region is defined as: $ R= \{0\le x \le 1, y\ge0, z\ge0, 0\le y+z\le1 \}$
The process of calculating multivariable integrals by itself is not that much complicated, but I do not know how to start or how to "write down" the integral.
Here is my attempt:
$\int_0^1 \int_0^{1-z} \int_0^{1-y} (x*y*z) dz dy dx$
but I am not sure about it, as solving this won't give me a "number" in the end, but an expression with z in it.
You're almost there, but the integratin domain is not quite correct, in the end no variables should be left as you correctly determined, and that is because of the superfluous $z$ in the bounds of $y$, the correct way would be following:
$$\int_0^1 \int_0^1 \int_0^{1-y} f(x,y,z)dz dy dx$$
Now we have $z \in [0,1-y]$ and $y \in [0,1]$, so certianly $0\leq y+z$ but because $z\leq 1-y$ we get $y+z \leq y + 1-z =1$.