Fubini's theorem

Fubini's theorem is the permission slip that turns a double or triple integral — defined abstractly as a limit over tiles — into something you can actually compute: a stack of ordinary single integrals done one after another. It says that the genuine integral over a region equals the iterated integral, and moreover that you may integrate the variables in either order and get the same number. Without it, the iterated integral would just be a hopeful recipe with no guarantee it matches the thing you want.

Stated for a double integral: if f(x, y) is reasonably well-behaved on a region R, then the double integral over R of f dA equals integral over x of [integral over y of f dy] dx, and equals integral over y of [integral over x of f dx] dy. The crucial fine print is what 'well-behaved' means. The safe sufficient condition (sometimes called Tonelli's part) is that f is continuous, or more generally that the integral of the absolute value of f is finite — that f is absolutely integrable over R. Under that condition the order does not matter. The cleanest application is on a rectangle, where the inner and outer limits are all constants and the two orders are obviously symmetric.

The honest warning is that the absolute-integrability hypothesis is real, not decorative. There are classic counterexamples — functions on the unit square, or sums that depend on order — where switching the order of integration changes the answer because the positive and negative parts each integrate to infinity and the cancellation is order-dependent. So when you reverse the order of integration to make a stubborn integral tractable (a standard and powerful move), you are implicitly invoking Fubini, and you should be sure its hypotheses hold, especially over unbounded regions or with sign-changing integrands.