{"id":5596,"date":"2014-07-28T06:05:25","date_gmt":"2014-07-27T21:05:25","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5596"},"modified":"2014-08-05T08:23:09","modified_gmt":"2014-08-04T23:23:09","slug":"triple-integrals","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5596","title":{"rendered":"TRIPLE INTEGRALS"},"content":{"rendered":"

\n\"Fig6-x\"<\/a><\/p>\n

The above results are easily generalized to closed regions in three dimensions. For example, consider a function F(x, y, z) defined in a closed three dimensional region \\cal R. Subdivided the region into n<\/em> subregions of volume \\Delta V_k,\\ k = 1,\\ 2,\\ \\dots,\\ n. Letting (\\xi_k, \\eta_k, \\zeta_k) be some point in each subregion, we form <\/p>\n

\\displaystyle \\lim\\limits_{n\\rightarrow\\infty}\\sum_{k=1}^{n}F(\\xi_k, \\eta_k, \\zeta_k)\\Delta V_k\\cdots(6)<\/p>\n

where the number n<\/em> of subdivisions approaches infinity in such a way that the largest linear dimension of each subregion approaches zero. If this limit exists we denote it by<\/p>\n

\\displaystyle \\underset{\\cal R}{\\iiint}F(x, y, z)dV\\cdots(7)<\/p>\n

called the triple integral<\/em> of F(x, y, z) over \\cal R. The limit dose exist if F(x, y, z) is continuous (or piecewise continuous) in \\cal R. <\/p>\n

If we construct a grid consisting of planes parallel to the xy<\/em>, yz<\/em> and xz<\/em> planes, the region \\cal R is subdivided into subregions which are rectangular parallelepipeds. In such case we can express the triple integral over \\cal R given by (7) as an iterated integral<\/em> of the form<\/p>\n

\\displaystyle \\int_{x=a}^{b}\\int_{y=g_1(x)}^{g_2(x)}\\int_{z=f_1(x,y)}^{f_2(x,y)}F(x, y, z)dxdydz = \\\\ \\int_{x=a}^{b} \\left [ \\int_{y=g_1(x)}^{g_2(x)} \\left \\{ \\int_{z=f_1(x,y)}^{f_2(x,y)} F(x, y, z)dz \\right \\} dy \\right ] dx\\cdots(8)<\/p>\n

(where the innermost integral is to be evaluated first) or the sum of such integrals. The integration can also be performed in any other order to give an equivalent result. <\/p>\n

Extensions to higher dimensions are also possible. <\/p>\n<\/blockquote>\n