{"id":5188,"date":"2021-09-02T00:58:06","date_gmt":"2021-09-01T19:28:06","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=5188"},"modified":"2026-02-08T13:33:10","modified_gmt":"2026-02-08T08:03:10","slug":"what-is-numerical-integration","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-numerical-integration\/","title":{"rendered":"What is numerical integration?"},"content":{"rendered":"\n

Overview<\/strong> (numerical integration)<\/h2>\n\n\n\n

Numerical integration<\/a> is the method to calculate the approximate value of the integral by using numerical techniques.<\/p>\n\n\n\n

There are various useful and interesting methods for numerical integration<\/a> such as trapezoidal rule, Simpson\u2019s rules, Gauss\u2019s, Newton-Leibnitz rules etc.<\/p>\n\n\n\n

Generally, we use fundamental theorem of calculus to find the solution for definite integrals.<\/p>\n\n\n\n

Sometime integration becomes too hard to evaluate therefore, numerical methods are used to find the approximated value of the integral.<\/p>\n\n\n\n

Numerical Integration<\/strong><\/h2>\n\n\n\n

Numerical integration<\/a> is a process of evaluating or obtaining a definite integral from a set of numerical values of the integrand f(x).<\/p>\n\n\n\n

In case of function of single variable, we call this as quadrature.<\/p>\n\n\n\n

Trapezoidal Method<\/strong><\/h2>\n\n\n\n

Suppose, the interval [a,b] be divided into n equal intervals such that <\/p>\n\n\n\n

Here <\/p>\n\n\n\n

To find the value of  <\/p>\n\n\n\n

Setting n=1, we get<\/p>\n\n\n\n

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

Or<\/p>\n\n\n\n

I = \u00bd (h) [(sum of the first and last ordinates) + 2( sum of all intermediate ordinates)]<\/p>\n\n\n\n

The above is the Trapezoidal rule.<\/p>\n\n\n\n

Note:<\/strong> In this method second and higher difference has to be neglected and so that f(x) is a polynomial of degree 1. <\/p>\n\n\n\n

Geometrical Significance:<\/strong> The curve y=f(x), we replace by n straight lines with the points <\/p>\n\n\n\n

The area bounded by the curve y = f(x), the ordinates x=xo and x=xn the x axis is approximately equivalent to the sum of the area of the n trapeziums obtained.<\/p>\n\n\n\n

Example: Solve using trapezoidal rule with five ordinates<\/strong><\/p>\n\n\n

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

Sol:<\/p>\n\n\n\n

Here, we have<\/p>\n\n\n

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

Now, we construct the data table:<\/p>\n\n\n\n

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

I = \u00bd (h) [(sum of the first and last ordinates) + 2( sum of all intermediate ordinates)]<\/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 technique 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

Therefore we will study about Simpson\u2019s 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 quarature formula-<\/p>\n\n\n\n

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

In this paragraph, we are going to discuss two very important methods, Simpson’s one-third rule and Simpson\u2019s three-eight rules.<\/a><\/p>\n\n\n\n

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

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

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

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

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

Simpson\u2019s one-third rule<\/strong><\/h2>\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>\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><\/h2>\n\n\n\n

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

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

\"\"\/<\/figure>\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 of numerical integration<\/h2>\n\n\n\n

Example: Calculate the value the following integral by<\/strong> using the<\/strong> Simpson\u2019s rules<\/strong><\/p>\n\n\n\n

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

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

First of all we will divide the interval into six part, where width (h) = 1, the value of f(x) will be<\/p>\n\n\n\n

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

Now we will use Simpson\u2019s 1\/3rd<\/sup> rule-<\/p>\n\n\n\n

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

Therefore we get<\/p>\n\n\n\n

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

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

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

The table of the values, we get<\/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\n\n\n

Interested in learning about similar topics? Here are a few hand-picked blogs for you!<\/p>\n\n\n\n