How do I find a region bound by three planes and a parabolic cylinder?

Question

Find the volume of the solid B bounded by the parabolic cylinder $x = (y − 4)^2 + 3$ and the planes $z = x + 2y − 4$, $z = x + 4y − 7$ and $x + 2y = 11$

I tried approaching by first looking at the x-z axis, and got:

$11 \leq x \leq 19$ and $ x-7 \leq z \leq x-4 $, but I am certain this approach is wrong and I can't find how to get the bounds for y anyway.

Any hints?

Answer 1

Hint:

Let $X=x-3$, $Y=y-4$ and $Z=z-9$, then with this coordinate translation the problem is

Find the volume of the solid $B$ bounded by the parabolic cylinder $X=Y^2$ and the planes $Z=X+2Y−2$, $Z=X+4Y+3$ and $X+2Y=0$.