I've tackled quite a few questions involving changing the order of integration over a general regions, however, for some reason, this one is troubling me. I am supposed to find three different iterated integrals to represent the given integral over the region specified, but after getting the first two, I can't seem to figure out how to structure the third. Consider the problem:
Using Fubini's theorem, write the three iterated version of the integral of the function $f(x,y,z) = xyz$ over the region bounded by $x, y, z\geq 0$, $x+2y-3\leq 1, z\leq x+y$ (you do not have to solve them).
I got: $$\int_{0}^{2}\int_0^{4-2y}\int_{0}^{x+y} xyz ~dz ~dx ~dy$$ and $$\int_{0}^{4}\int_0^{2-x/2}\int_{0}^{x+y} xyz ~dz ~dy ~dx$$
Now, the last one cannot have $z$ as the first thing to be integrated, as we've already seen both of those possibilities. However, I am unsure how to write $z$ in terms of solely $x$ or $y$. Once I do that, how do I write (wlog) $x$ in terms of $x$ and $z$?
After a bit of digging, I found that $0\le z \le 4-y$, which may be useful. However, now I need to write $x$ in terms of $y,z$, which has proved to be a challenge. Thanks for any help!