# Iterated Integral

## iterated integral

[′īd·ə‚rād·əd ′int·ə·grəl]*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## Iterated Integral

a concept of the integral calculus. Let

*I* = ∫∫_{S}*f(x, y) dxdy*

be the double integral of the function *f(x, y)* over the region *S* bounded by the lines *x* = *a* and *x = b* and the curves *y* = φ_{1}*(x)* and *y* = φ_{2}*(x)*. If certain conditions on *f(x, y)*, φ_{1}*(x)*, and φ_{2}*(x)* are fulfilled, the double integral can be calculated by the formula

where *x* is kept constant when the inner integral is calculated. The calculation of a double integral thus reduces to two calculations of ordinary integrals, or to the calculation of what is called an iterated integral.

Geometrically, the reduction of a double integral to an iterated integral means that the volume of a cylindroid can be calculated both by dividing it into elementary columns and by dividing it into elementary layers parallel to the *yz*-plane. The order of integration in the iterated integral can be changed—that is, integration may be performed first with respect to *x* and then with respect to *y*—if certain conditions are imposed on *f(x, y)* and *S*. The iterated integral is defined in a similar manner for functions of more than two variables.