We can calculate the value of a definite integral by using trapezoids to divide the area under the curve for the given function.<\/p>\n\n\n\n
Trapezoidal Rule Statement:<\/strong> Let f(x) be a continuous function on the interval (a, b). Now divide the intervals (a, b) into n equal sub-intervals with each of width,<\/p>\n\n\n\n \u0394x = (b – a)\/n<\/strong>, such that a = x0<\/sub> < x1 <\/sub>< x2 <\/sub>< x3 <\/sub><\u2026..< xn<\/sub> = b<\/p>\n\n\n\n Then the Trapezoidal Rule formula for area approximating the definite integral b<\/sup>\u222ba<\/sub>f(x)dx is given by:<\/p>\n\n\n\n b<\/sup>\u222ba<\/sub>f(x) dx \u2248 Tn<\/sub> = \u25b3x\/2 [f(x0<\/sub>) + 2f(x1<\/sub>) + 2f(x2<\/sub>) +\u2026.2f(xn-1<\/sub>) + f(xn<\/sub>)]<\/p>\n\n\n\n where, xi<\/sub> = a + i\u25b3x<\/p>\n\n\n\n If n \u2192 \u221e, R.H.S of the expression approaches the definite integral b<\/sup>\u222ba<\/sub> f(x)dx<\/p>\n\n\n\n Proof:<\/strong><\/p>\n\n\n\n To prove the trapezoidal rule, consider a curve as shown in the figure above and divide the area under that curve into trapezoids. We see that the first trapezoid has a height \u0394x and parallel bases of length y0<\/sub> or f(x0<\/sub>) and y1<\/sub> or f1<\/sub>. Thus, the area of the first trapezoid in the above figure can be given as,<\/p>\n\n\n\n (1\/2) \u0394x [f(x0<\/sub>) + f(x1<\/sub>)]<\/p>\n\n\n\n The areas of the remaining trapezoids are (1\/2)\u0394x [f(x1<\/sub>) + f(x2<\/sub>)], (1\/2)\u0394x [f(x2<\/sub>) + f(x3<\/sub>)], and so on.<\/p>\n\n\n\n Consequently,<\/p>\n\n\n\n \u222bb<\/sup>a <\/sub>f(x) dx \u2248 (1\/2)\u0394x (f(x0<\/sub>)+f(x1<\/sub>) ) + (1\/2)\u0394x (f(x1<\/sub>)+f(x2<\/sub>) ) + (1\/2)\u0394x (f(x2<\/sub>)+f(x3<\/sub>) ) + \u2026 + (1\/2)\u0394x (f(n-1<\/sub>) + f(xn<\/sub>) )<\/p>\n\n\n\n After taking out a common factor of (1\/2)\u0394x and combining like terms, we have,<\/p>\n\n\n\n \u222bb<\/sup>a <\/sub>f(x) dx\u2248 (\u0394x\/2) (f(x0<\/sub>)+2 f(x1<\/sub>)+2 f(x2<\/sub>)+2 f(x3<\/sub>)+ … +2f(n-1<\/sub>) + f(xn<\/sub>) )<\/p>\n","protected":false},"excerpt":{"rendered":" We can calculate the value of a definite integral by using trapezoids to divide the area under the curve for the given function. Trapezoidal Rule Statement: Let f(x) be a continuous function on the interval (a, b). Now divide the intervals (a, b) into n equal sub-intervals with each of width, \u0394x = (b – a)\/n, […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[874],"tags":[],"_links":{"self":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts\/4394"}],"collection":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/comments?post=4394"}],"version-history":[{"count":0,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts\/4394\/revisions"}],"wp:attachment":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/media?parent=4394"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/categories?post=4394"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/tags?post=4394"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}