What are Simpson’s rules for numerical integration?

Overview

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.

Simpson’s rules are very useful in numerical integration to evaluate such integrals.

Here we will understand the concept of Simpson’s rule and evaluate integrals by using numerical techniques of integration.

We find more accurate value of the integration by using Simpson’s rule than other methods

Simpson’s rule

We will study Simpson’s one-third rule and Simpson’s three-eight rules.

But in order to get these two formulas, we should have to know about the general quadrature formula-

General quadrature formula-

The general quadrature formula is gives as-

Simpson’s one-third and three-eighth formulas are derived by putting n = 2 and n = 3 respectively in the general quadrature formula.

Simpson’s one-third rule-

 Put n = 2 in general quadrature formula-

We get-

Note- the given interval of integration has to be divided into an even number of sub-intervals.

Simpson’s three-eighth rule-

 Put n = 3 in general quadrature formula-

We get-

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

Solved examples

Example: Evaluate the following integral by using Simpson’s 1/3^rd and 3/8^th rule.

Solution-

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

Now using Simpson’s 1/3^rd rule-

We get-

And now

Now using Simpson’s 3/8^th rule-

Example: Find the approximated value of the following integral by using Simpson’1/3^rd rule.

Solution-

The table of the values-

Now using Simpson’s 1/3^rd rule-

We get-