Fubini’s theorem Let $I\subset\mathbb{R}^{N}$ and $J\subset\mathbb{R}^{M}$ be compact intervals, and let $f:I\times J\to\mathbb{R}^{K}$ be a Riemann integrable function such that, for each $x\in I$ the integral

exists. Then $F:I\to\mathbb{R}^{K}$ is Riemann integrable, and

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 $f:I\to\mathbb{R},x\mapsto\sin(x)\cos(y)$ be a function. Then

Note that it is often simpler (and no less correct) to write $\idotsint_{I}f$ as $\int_{I}f$.