I have several tasks to solve where a set of inequalities is used to describe a region. I should then calculate the area or volume of that region.
Let's say we have the following inequality (for $x,y,z \geq 0$):
$x+2y+3z \leq 1$
Now I need to find out the boundaries for the triple integral. From a few examples I have here, I wasn't able to derive a way to solve such a problem.
The example solution suggests that
$0 \leq y \leq \frac{1-x}{2}$
and
$0 \leq z \leq \frac{(x - 2y)}{3}$.
for
$0 \leq x \leq 1$
leading to the integral:
$ \int_{0}^{1} \int_{0}^{\frac{1-x}{2}} \int_{0}^{\frac{x-2y}{3}} { 1 \; dz dy dx } $
How do I find out these boundaries? (Moreover, how do I find out the boundaries in a general approach)
PS: I'm not sure if the solution is correct at all, that's probably why I'm confused.