The following theorem allows one to compute the integral of an integrable multivariable function using iterated integration (i.e. integration sections of the function). # Statements > [!NOTE] Fubini's theorem for Riemann integration > Let $A\subset \mathbb{R}^{m}$, $B\subset \mathbb{R}^{n}$ be rectangles (products of closed intervals), and $f:A\times B \to \mathbb{R}$ be [[Riemann Integration|Riemann integrable]]. For fixed $x\in A$, define $g_{x}: B \to \mathbb{R}$ by $g_x(y)=f(x,y)$ (known as the section, or slice, of $f$ at $x$) and denote its [[Riemann Integration|upper and lower Riemann sums]] by $\begin{align} > \mathcal{L}(x) &= \mathbf{L}\int_{B} g_{x} = \mathbf{L}\int_{B} f(x,y) \, dy, \\ > \mathcal{U}(x) &= \mathbf{U}\int_{B} g_{x} = \mathbf{U} \int_{B} f(x,y) \, dy. > \end{align}$Then $\mathcal{L}$ and $\mathcal{U}$ are Riemann integrable on $A$ and $\begin{align} \int_{A\times B} f &= \int_{A} \mathcal{L} = \int_{A} \left( \mathbf{L} \int_{B} f(x,y) \, dy \right) \, dx \\ \int_{A\times B} f &= \int_{A} \mathcal{U} = \int_{A} \left( \mathbf{U} \int_{B} f(x,y) \, dy \right) \, dx.\end{align}$ ###### Remarks 1. If for all $x\in A$, $g_{x}$ is Riemann integrable, then $\mathcal{L}(x) =\int_{B}f(x,y) \,dy= \mathcal{U}(x)$ which yields $\int_{A\times B} f = \int_{A} \left( \int_{B} f(x,y) \, dy \right) \, dx.$This is certainly the case if $f$ is continuous (see [[Continuous Functions are Separately Continuous]] and [[Continuous real functions are Riemann Integrable]]). # Proofs