{"id":5572,"date":"2014-07-14T06:05:30","date_gmt":"2014-07-13T21:05:30","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5572"},"modified":"2014-08-05T08:26:08","modified_gmt":"2014-08-04T23:26:08","slug":"double-integrals","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5572","title":{"rendered":"DOUBLE INTEGRALS"},"content":{"rendered":"

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

Let F(x, y) be defined in a closed region \\cal R of the xy plane. Subdivided \\cal R into n subregions \\Delta\\cal R of area \\Delta A_k,\\ k = 1,\\ 2,\\ \\dots,\\ n. Let (\\xi_k, \\eta_k) be some point of \\Delta\\cal R. Form the sum<\/p>\n

\\displaystyle \\sum_{k=1}^{n}F(\\xi_k, \\eta_k)\\Delta A_k\\cdots(1)<\/p>\n

Consider<\/p>\n

\\displaystyle \\lim\\limits_{n\\rightarrow\\infty}\\sum^{n}_{k=1}F(\\xi_k, \\eta_k)\\Delta A_k\\cdots(2)<\/p>\n

where the limit is taken so that the number n<\/em> of subdivisions increases without limit and such that the largest linear dimension of each \\Delta \\cal R approaches zero. If this limit exists it is denoted by<\/p>\n

\\displaystyle \\iint_{\\cal R}F(x, y)dA\\cdots(3)<\/p>\n

and is called the double integral<\/em> of F(x, y)<\/em> over the region \\cal R. <\/p>\n

It can be proved that the limit dose exist if F(x, y) is continuous (or piecewise continuous) in \\cal R. <\/p>\n<\/blockquote>\n