{"id":5583,"date":"2014-07-21T06:05:04","date_gmt":"2014-07-20T21:05:04","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5583"},"modified":"2014-08-05T08:25:06","modified_gmt":"2014-08-04T23:25:06","slug":"iterated-integrals","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5583","title":{"rendered":"ITERATED INTEGRALS"},"content":{"rendered":"

\n\"Fig.<\/a><\/p>\n

If \\cal R is such that any lines parallel to the y<\/em> axis meet the boundary of \\cal R in at most two points, then we can write the equation of the curves ACB<\/em> and ADB<\/em> 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<\/em> and y<\/em> axes and \\Delta A_k as the corresponding areas. Then (3) can be written<\/p>\n

\\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)<\/p>\n

where the integral in braces is to be evaluated first (keeping x<\/em> constant) and finally integrating with respect to x<\/em> from a<\/em> to b<\/em>. The result (4) indicates how a double integral can be evaluated by expressing it in terms of two single integrals called iterated integrals<\/em>.<\/p>\n

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

\\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)<\/p>\n

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<\/em> with respect to the other form. <\/p>\n

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. <\/p>\n<\/blockquote>\n