order of integration

When you write a double integral as an iterated integral, you must decide which variable to integrate first — do you sweep horizontally then vertically, or the other way around? That choice is the order of integration. For a rectangle it makes no difference to the difficulty, but over a curved or slanted region one order can be a single clean integral while the other forces you to split the region into pieces or wrestle with an integrand that has no elementary antiderivative. Knowing how to switch from one order to the other is one of the most useful manual skills in multivariable calculus.

Reversing the order is not just swapping dx dy for dy dx — that alone gives the wrong answer. You must re-derive the limits from the geometry of the region. The standard procedure: sketch the region R; for the order dy dx, fix a typical x and read off how y ranges (between a lower curve and an upper curve), then let x run between its extreme constant values; for dx dy, fix a typical y and read off how x ranges, then let y run between its constants. The same region, described two ways, yields two iterated integrals that Fubini's theorem certifies are equal. A region that needs two pieces in one order may need only one in the other.

This is far more than tidiness. The classic payoff is an integral like integral from 0 to 1 of [integral from x to 1 of e^{-y^2} dy] dx, which is hopeless as written because e^{-y^2} has no elementary antiderivative — but reverse the order to dx dy and the inner x-integral becomes trivial, collapsing the whole thing to something elementary. Swapping the order is also how you tame integrals over triangles, regions between intersecting curves, and many improper integrals. The single most common mistake is changing the differentials but copying the old numeric limits; the limits must be rebuilt from the picture every time.