Simpsons Rule Formula

Simpsons Rule Formula

One of the Simpsons Rule Formula used to get the approximate value of a definite integral is Simpson’s rule. An integral possessing lower and upper boundaries are referred to as a defined integral. In order to assess a definite integral, typically first integrate (using integration techniques), and then apply the limits using the calculus fundamental theorem. But occasionally, people are unable to calculate an integral using any integration approach, and occasionally, they do not have a precise function to integrate; instead, we have some observed values of the function (in the case of experiments). In these circumstances, Simpson’s rule aids in estimating the definite integral’s value.

What is Simpson’s Rule?

Simpson’s rule is used to approximate the area under the graph of the function f to get the value of a definite integral (that is, of the form b∫ₐ f(x) dx) (x). When using Simpson’s method, assess the area under a curve by dividing the whole area into parabolas, however, when using the Riemann sum, evaluate the area under a curve (a definite integral) by dividing the area under the curve into rectangles. The Simpson’s 1/3 rule, or Simpson’s one-third rule, is another name for the Simpson’s rule.

Simpson’s Rule Formula

There are a number of numerical techniques that can be used to estimate an integral, including Riemann’s left and right sums, the midpoint rule, the trapezoidal rule, the Simpsons Rule Formula etc. However, the Simpsons Rule Formula provides the closest approximation of a definite integral among these. The Simpsons Rule Formula is: If f(x) = y is evenly distributed between [a, b], then we have:

b∫a f(x) d x ≈ (h/3) [f(x0)+4 f(x1)+2 f(x2)+ … +2 f(xn-2)+4 f(xn-1)+f(xn)]


The interval [a, b] should be broken into n subintervals, where n is an even number.

(The problem typically includes the letter n)

h = [(b – a) / n] when x0 = a and xn = b

The ends of the n subintervals are x0, x1,…., and xn.

Simpson’s Rule Error Bound

Simpson’s approach only provides a rough estimate of the integral’s value, rather than its precise value. Therefore, an error is always there and may be determined using the method below.

Simpson’s rule’s error bound is m(b-a)5/180n4.

Simpson’s 1/3 Rule Derivation

To approximate the value of the definite integral ba f(x) dx, let’s use Simpson’s 1/3 rule, which divides the area under the curve f(x) into parabolas. For this, let’s split the range [a, b] into n subranges [x0, x1], [x1, x2], [x2, x3], …, [xn-2, xn-1], [xn-1, xn] each having width ‘h’, where x0 = a and xn = b.

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How to Apply Simpson’s Rule?

Simpson’s 1/3 rule provides a closer approximation. The steps for applying Simpson’s rule to approximate the integral ba f(x) dx are listed below.

Step 1: Determine the values of “a” and “b” from the interval “a, b,” as well as the value of “n,” which represents the number of subintervals.

Step 2: Determine each subinterval’s width using the equation h = (b – a)/n.

Step 3: Using the interval width ‘h,’ divide the interval [a, b] into ‘n’ subintervals [x0, x1], [x1, x2], [x2, x3], …, [xn-2, xn-1], [xn-1, xn].

Step 4: Simplify the Simpsons Rule Formula by substituting all of these values.

b a f (x) dx h / 3 The expression is [f(x0)+4 f(x1)+2 f(x2)+… +2 f(xn-2)+4 f(xn-1)+f(xn)].

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FAQs (Frequently Asked Questions)

1. Why does Simpson's rule not contain a mistake?

However, cubic polynomials also follow Simpson’s Rule precisely. Since it is an absolute value and the error is less than or equal to 0, it must be 0. As a result, cubic polynomials always follow Simpson’s Rule exactly.

2. How does Simpson's rule define K?

This means that K must always be greater than or equal to the second derivative of the given function for the midpoint and trapezoidal rules and that K must always be greater than or equal to the fourth derivative of the given function for the Simpson’s rule.