I'm trying to set up all six triple integrals to find the volume of the solid that lies in the first octant bounded by the coordinate planes, the plane $y + x = 4$, and the cylinder $16 - 4z^2 = y^2$.
$3$D graph
I've been able to set up the triple integrals for every other combination except for $dydxdz$ and $dydzdx$, which I am struggling to find the bounds for. I know the volume is $8\pi - 32/3$, but I keep getting an incorrect answer when I set up and calculate dydxdz and dydzdx.
Here's my equation for the projection onto the $xz$-plane, but I'm not sure it is correct/complete:
$xz$-plane projection?
I'd appreciate any help regarding how to find the bounds of dydxdz and dydzdx.