Iterated Integral

a concept of the integral calculus. Let

I = ∫∫_Sf(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 = φ₁(x) and y = φ₂(x). If certain conditions on f(x, y), φ₁(x), and φ₂(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.