Fubini’s theorem

Fubini’s theoremLet $I\subset {ℝ}^N$ 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

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