Stack boxes under a surface until they stop being boxes. That limit is the double integral.
The midpoint rule is far better than either corner, and for a reason worth thinking about.
Raise the box count and watch the total settle.
The double integral is defined as a limit of exactly this picture. Chop the region into little rectangles, pick a point in each, build a column of that height, and add up the volumes. Then let the rectangles shrink. Everything else — iterated integrals, changing the order, Fubini's theorem — is machinery for computing this number without ever building the boxes.
Which sample point you pick inside each cell doesn't matter in the limit; that's part of what it means for the integral to exist. But it matters a great deal at any finite N. Sampling the lower-left corner systematically undershoots a rising surface and the upper-right systematically overshoots, so both errors shrink only as fast as 1/N.
The midpoint rule does much better — its error falls like 1/N² — because the error it makes on the low side of each cell very nearly cancels the error on the high side. It's the same reason the midpoint rule beats left endpoints in one variable, and it's why nobody actually computes integrals with corners.
Watch the boxes as N grows and the staircase becomes a surface. That transition is the whole content of the definition.