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