Fubini's theorem

A theorem guaranteeing a double integral over a rectangle equals either iterated order, and both orders give the same number.

Fubini's theorem rescues the double integral from Riemann-sum tedium: over a rectangle, it collapses into two ordinary single integrals, done one at a time, with every antiderivative technique already available. Because both orders give the same number for continuous $f$ on a rectangle, picking the order that produces the easier antiderivative is always legitimate — nothing about the theorem requires matching the order a problem happened to state the region in.

Examples

$\int_0^2\!\int_0^1 xy^2\,dy\,dx=\tfrac23$ , and swapping the order to $\int_0^1\!\int_0^2 xy^2\,dx\,dy$ gives the same $\tfrac23$ .