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What are Simpson’s rules for numerical integration?

Overview(Simpson’s rules)

Generally we use the fundamental theorem of calculus 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 rules

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-

The general quadrature formula is gives as-

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

One-third rule

Put n = 2 in general quadrature formula-

We get-

Three-eighth rule

Put n = 3 in general quadrature formula-

We get-

Solved examples

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

Solution-

First we 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 1/3rd rule-

We get-

And now

Now using Simpson’s 3/8th rule-

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

Solution-

The table of the values-

Now using Simpson’s 1/3rd rule-

We get-

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