{"id":5671,"date":"2014-08-25T06:05:53","date_gmt":"2014-08-24T21:05:53","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5671"},"modified":"2014-08-08T19:47:28","modified_gmt":"2014-08-08T10:47:28","slug":"evaluation-of-line-integrals","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5671","title":{"rendered":"EVALUATION OF LINE INTEGRALS"},"content":{"rendered":"
\"Fig.<\/a>
Fig. 6-2<\/figcaption><\/figure>\n

If the equation of a curve C<\/em> in the plane z = 0 is given as y = f(x), the line integral (14) is evaluated by placing y = f(x),\\ dy = f'(x)dx in the integrand to obtain the definite integral <\/p>\n

\\displaystyle \\int_{a_1}^{a_2}[P\\{x, f(x)\\}dx + Q\\{x, f(x)\\}f'(x)dx] \\cdots(19)<\/p>\n

which is then evaluated in the usual manner. <\/p>\n

Similarly if C<\/em> is given as x = g(y), then dx = g'(y)dy and the line integral becomes <\/p>\n

\\displaystyle \\int_{b_1}^{b_2}[P\\{g(y), y\\}g'(y)dy + Q\\{g(y), y\\}dy]\\cdots(20)<\/p>\n

If C<\/em> is given in parametric form x = \\phi(t),\\ y = \\psi(t), the line integral becomes <\/p>\n

\\displaystyle \\int_{t_1}^{t_2} [P\\{ \\phi(t),\\ \\psi(t) \\}\\phi'(t)dt + Q\\{ \\phi(t),\\ \\psi(t) \\}\\psi'(t)dt] \\cdots (21)<\/p>\n

where t_1 and t_2 denote the values of t corresponding to points A and B respectively. <\/p>\n

Combination of the above methods may be used in the evaluation. <\/p>\n

Similar methods are used for evaluating line integrals along space curve. <\/p>\n<\/blockquote>\n