I am going through a practice exam for my calculus course and I found this question :
Evaluate $\iiint_{G} xy\ d V$ where $G$ is the region bounded by $$x+y+z \leq 1, \quad y \leq x, \quad x \geq 0, \quad y \geq 0, \quad z \geq 0$$
The answer given straight away says $$\int_{0}^{0.5} \int_{x=y}^{x=1-y} \int_{0}^{z=1-x-y} x y d z d x d y$$ without calculating the domain, but it does not seem too obvious to me how it is done. After working through algebra, I can see that you can find $y>0.5$ using the inequalities but by the same logic can't I show $x>1$ instead of $x>1-y$ ? How would I go about intuitively finding these limits? Can it only be done by finding all the intersection points?
You need to sketch the region of integration to see what is going on with the limits.
On the $xy$ plane you have the base which is the triangular region between the $x$ axis, the lines $y=x$ and $y=1-x$
The ceiling is the plane $z=1-x-y$ and the floor is the $xy$ plane which is $z=0$
That should have clarified the limits once you have a three dimensional graph to look at it.