A solid casts three shadows, and each one is the outer region of a different order of integration.
The shadow is not decoration. It is literally the region the outer double integral is taken over.
Switch the order and watch which shadow becomes the floor.
Setting up a triple integral is a question about shadows. Whichever variable you integrate first, you're firing a needle through the solid in that direction; the inner limits are where the needle enters and leaves. Once every needle in a given line of fire is accounted for, what remains is a flat region — the solid's shadow on the perpendicular plane — and that's what the outer double integral runs over.
So the recipe is mechanical once you see it. Pick a direction. Find where a needle along it enters and exits: those are your inner limits, and they are functions of the other two variables. Then project the solid onto the plane perpendicular to that direction and set up an ordinary double integral over the projection.
Switch the order and every part of that changes. Different needle, different entry and exit surfaces, different shadow. The solid is the same, the volume is the same, and the three sets of limits look nothing alike — which is why "just draw it" is the standing advice, and why the hard part of a triple integral is almost never the antiderivative.
Some orders are much kinder than others. Watch which shadow is a simple shape and which is awkward, and you have your answer about which order to actually use.