Here's a haphazard attempt at a graph of your region. I hope it gives you some idea of what it looks like.
Because that's where the plane 'slices' the parabola x^2.
Here's a graph. The line y=4 is where the plane z=4-y meets the parabola.
So, you integrate over y from x^2 to 4. See?.
At x=2, y=x^2=4
The 3-D I posted is rather cock-eyed, but you can see it there.
I gonna go mow grass now, so I may not be back for a few hours.
First, try drawing your parabola y=x^2. That's easy enough. It's the plane rising up the z-axis that's the booger to visualize. Try plotting various points for it.
z=4-y, if y=0, z=4
If y=4, z=0. See there?. That's where the plane intersects the parabola in the xy plane.
Then it rises up the z-axis at a slant and intersects the z axis at y=0.