If $f\left(x\,y\right)$ is continuous over the rectangle $R$ defined by $a\le x\le b\,c\le y\le d$, its double integral over $R$ can be evaluated by either of the two iterated integrals

$\int {\int }_{R}f\left(x\,y\right) \mathrm{dA}$ = ${\int }_a^b{\int }_c^{d}f\left(x\,y\right) \mathrm{dy} \mathrm{dx}$ = ${\int }_c^d{\int }_a^{b}f\left(x\,y\right) \mathrm{dx} \mathrm{dy}$ 

There are weaker conditions under which the double integral of $f$ can be evaluated by iteration, but in a first course in multivariate calculus, continuity suffices.

The mechanics of the iterated integral are detailed in the following examples.