ITERATED INTEGRALS

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Fig. 6-1

If \cal R is such that any lines parallel to the y axis meet the boundary of \cal R in at most two points, then we can write the equation of the curves ACB and ADB bounding \cal R as  y = f_1(x) and  y = f_2(x) respectively, where f_1(x) and f_2(x) are single-valued and continuous in a \le x \le b. In this case we can evaluate the double integral (3) by choosing the regions \Delta \cal R as rectangles formed by constracting a grid of lines parallel to the x and y axes and \Delta A_k as the corresponding areas. Then (3) can be written

\displaystyle \iint_{\cal R} F(x, y)dxdy = \int_{x=a}^{b}\int_{y=f_1(x)}^{f_2(x)}F(x, y)dydx = \int_{x=a}^{b}\left\{\int_{y=f_1(x)}^{f_2(x)}F(x, y)dy\right\}dx\cdots(4)

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 \cal R is such that any lines parallel to the x axis meet the boundary of \cal R in at most two points, then the equations of curves CAD and CBD can be written x = g_1(y) and x = g_2(y) respectively and we find similarly

\displaystyle \iint_{\cal R}F(x,y)dxdy = \int_{y=c}^{d}\int_{x=g_1(y)}^{g_2(y)}F(x,y)dxdy = \int_{y=c}^{d}\left\{\int_{x=g_1(y)}^{g_2(y)}F(x,y)dx\right\}dy\cdots(5)

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 \cal R is not of the type shown in the above figure, it can generally be subdivided into regions {\cal R}_1,\ {\cal R}_2,\ \dots which are of this type. Then the double integral over \cal R is found by taking the sum of the double integrals over {\cal R}_1,\ {\cal R}_2,\ \dots.

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