5.4 Repeated integrals

Given a continuous function $f(x,y)$ on $R$, we can consider it as a function of the $x$ and $y$ variables separately. For each $y$ we can integrate in the $x$ variable to form $\int_{a}^{b}f(x,y)dx$; this expression gives a function of $y$, which we can integrate to form the repeated integral

Similarly, for each $x$, we can integrate in the $y$ variable to form $\int_{c}^{d}f(x,y)dy$, giving a function of $x$, and then integrate to form the ‘other’ repeated integral

The term iterated integrals is also commonly used.

When evaluating double integrals, it is important to sketch the region $R$ so as to get the limits the right way round. It is important to keep track of which variables are fixed and which are being integrated.