{"id":6476,"date":"2021-10-30T07:40:36","date_gmt":"2021-10-30T07:40:36","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=2921"},"modified":"2021-10-30T07:40:36","modified_gmt":"2021-10-30T07:40:36","slug":"what-are-simpsons-rules-for-numerical-integration-2","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-are-simpsons-rules-for-numerical-integration-2\/","title":{"rendered":"What are Simpson\u2019s rules for numerical integration?"},"content":{"rendered":"\n

Overview<\/strong><\/h2>\n\n\n\n

Generally the fundamental theorem of calculus is used to find the solution for definite integrals, but sometimes integration becomes too hard to evaluate, numerical methods are used to find the approximated value of the integral.<\/p>\n\n\n\n

Simpson\u2019s rules are very useful in numerical integration to evaluate such integrals.<\/p>\n\n\n\n

Here we will understand the concept of Simpson\u2019s rule and evaluate integrals by using numerical techniques of integration.<\/p>\n\n\n\n

We find more accurate value of the integration by using Simpson\u2019s rule than other methods<\/p>\n\n\n\n

Simpson\u2019s rule<\/strong><\/h2>\n\n\n\n

We will study Simpson’s one-third rule and Simpson\u2019s three-eight rules.<\/p>\n\n\n\n

But in order to get these two formulas, we should have to know about the general quadrature formula-<\/p>\n\n\n\n

General quadrature formula-<\/strong><\/p>\n\n\n\n

\"\"\/<\/figure><\/div>\n\n\n\n

The general quadrature formula is gives as-<\/p>\n\n\n\n

\"\"\/<\/figure><\/div>\n\n\n\n

Simpson\u2019s one-third and three-eighth formulas are derived by putting n = 2 and n = 3 respectively in the general quadrature formula.<\/p>\n\n\n\n

Simpson\u2019s one-third rule-<\/strong><\/p>\n\n\n\n

 <\/strong>Put n = 2 in general quadrature formula-<\/p>\n\n\n\n

We get-<\/p>\n\n\n\n

\"\"\/<\/figure><\/div>\n\n\n\n

Note- the given interval of integration has to be divided into an even number of sub-intervals.<\/p>\n\n\n\n

Simpson\u2019s three-eighth rule-<\/strong><\/p>\n\n\n\n

 <\/strong>Put n = 3 in general quadrature formula-<\/p>\n\n\n\n

We get-<\/p>\n\n\n\n

\"\"\/<\/figure><\/div>\n\n\n\n

Note- the given interval of integration has to be divided into sub-intervals whose number n is a multiple of 3.<\/p>\n\n\n\n

Solved examples<\/h2>\n\n\n\n

Example: Evaluate the following integral by using Simpson\u2019s 1\/3<\/strong>rd<\/sup><\/strong> and 3\/8<\/strong>th<\/sup><\/strong> rule.<\/strong><\/p>\n\n\n\n

\"\"\/<\/figure><\/div>\n\n\n\n

Solution-<\/strong><\/p>\n\n\n\n

First we will divide the interval into six part, where width (h) = 1, the value of f(x) are given in the table below-<\/p>\n\n\n\n

\"\"\/<\/figure><\/div>\n\n\n\n

Now using Simpson\u2019s 1\/3rd<\/sup> rule-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

We get-<\/p>\n\n\n\n

\"\"\/<\/figure><\/div>\n\n\n\n

And now<\/p>\n\n\n\n

Now using Simpson\u2019s 3\/8th<\/sup> rule-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Example: Find the approximated value of the following integral by using Simpson\u20191\/3<\/strong>rd<\/sup><\/strong> rule.<\/strong><\/p>\n\n\n\n

\"\"\/<\/figure><\/div>\n\n\n\n

Solution-<\/p>\n\n\n\n

The table of the values-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Now using Simpson\u2019s 1\/3rd<\/sup> rule-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

We get-<\/p>\n\n\n\n

\"\"\/<\/figure>\n","protected":false},"excerpt":{"rendered":"

Simpson\u2019s rules are very useful in numerical integration to evaluate such integrals.<\/p>\n","protected":false},"author":3,"featured_media":3483,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[31],"tags":[],"class_list":["post-6476","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-maths"],"yoast_head":"\nWhat are Simpson\u2019s rules for numerical integration? - Goseeko blog<\/title>\n<meta name=\"description\" content=\"Simpson\u2019s rules are very useful in numerical integration to evaluate such integrals.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.goseeko.com\/blog\/what-are-simpsons-rules-for-numerical-integration-2\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" 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