Fubini's theorem Let $I \subset \R^N$ and $J \subset \R^M$ be compact intervals, and let $f : I \times J \to \R^K$ be a Riemann integrable function such that, for each $x \in I$ the integral $$ F(x) := \int_J f(x, y)\, d\mu_J(y) $$ exists. Then $F:I\to\R^K$ is Riemann integrable, and $$ \int_I F = \int_{I\times J} f. $$

This theorem effectively states that, given a function of $N$ variables, you may integrate it one variable at a time, and that the order of integration does not affect the result.

Example Let $I := [0, \pi/2]\times[0,\pi/2]$ , and let $f : I \to \R, x \mapsto \sin(x)\cos(y)$ be a function. Then \begin{equation*} \begin{split} \int_I f &= \iint_{[0, \pi/2]\times[0,\pi/2]} \sin(x)\cos(y) \\ &=\int_0^{\pi/2} \left( \int_0^{\pi/2} \sin(x)\cos(y)\,dy\right)\,dx \\ &=\int_0^{\pi/2} \sin(x)\left(1 - 0\right)\,dx =(0 - -1) = 1. \end{split} \end{equation*} Note that it is often simpler (and no less correct) to write $\idotsint_I f$ as $\int_I f$ .