Fubini's theorem

The double integral of a continuous function of two variables can be computed via iterated integration, independent of the order: $$\int\int_{[a,b]\times [c,d]}f(x,y)dA=\int_a^b\int_c^d f(x,y)dydx = \int_c^d\int_a^b f(x,y)dxdy.$$

So, the operations of "partial definite integration" commute: $$ \newcommand{\ra}[1]{\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{cccccc} C([a,b]\times [c,d]) & \ra{\int_a^b(.)dx} & C([c,d]) \\ \da{\int_c^d(.)dy} & \searrow & \da{\int_c^d(.)dy} \\ C([a,b]) & \ra{\int_a^b(.)dx} & {\bf R} \end{array} $$