If is such that any lines parallel to the y axis meet the boundary of in at most two points, then we can write the equation of the curves ACB and ADB bounding as and respectively, where and are single-valued and continuous in . In this case we can evaluate the double integral (3) by choosing the regions as rectangles formed by constracting a grid of lines parallel to the x and y axes and as the corresponding areas. Then (3) can be written
where the integral in braces is to be evaluated first (keeping x constant) and finally integrating with respect to x from a to b. The result (4) indicates how a double integral can be evaluated by expressing it in terms of two single integrals called iterated integrals.
If is such that any lines parallel to the x axis meet the boundary of in at most two points, then the equations of curves CAD and CBD can be written and respectively and we find similarly
If the double integral exists, (4) and (5) will in general yield the same value. In writing a double integral, either of the forms (4) or (5), whichever is appropriate, may be used. We call one form an interchange of the order of integration with respect to the other form.
In case is not of the type shown in the above figure, it can generally be subdivided into regions which are of this type. Then the double integral over is found by taking the sum of the double integrals over .