Sweep a region with vertical strips, then with horizontal ones. Same region, two very different sets of limits.
Try the lens. Sweeping sideways runs into a corner where the region changes shape, and one integral is no longer enough.
Switch the order and watch the limits rewrite themselves.
A double integral is a total. The region says which points to add up; it says nothing about the order you visit them in. Both sweeps below cover exactly the same points and give exactly the same answer.
What changes is the bookkeeping. Sweeping with vertical strips, the outer variable is x and the inner limits are the top and bottom of the strip standing at that x — so the inner limits are functions of x. Sweeping with horizontal strips, everything swaps.
That's the whole reason to draw the region before setting up the integral. The limits are not something you can read off the integrand; they are a fact about the picture, and they are different facts depending on which way you slice.
Sometimes one order is simply harder. On the lens region, a horizontal strip is bounded by the lower parabola until y reaches 1, and by the upper one after that — so that order takes two integrals while the vertical order takes one. Choosing the order well is most of the skill.