How Simpson’s 1/3 rule differs from Trapezoidal rule? Drive the formula for Simson’s 1/3 rule.

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Difference Between Trapezoidal and Simpson Rule:

The trapezoidal rule is based on the Newton-Cotes formula. The trapezoidal rule works by approximating the region under the graph of the function f(x) as a trapezoidal and calculate its area. The trapezoidal rule assumes n = 1. That is, it approximates the integral by a linear polynomial (straight line).

\(\int_{x_0}^{x_1}f(x) dx = (x_1 – x_0)\left [ \frac{f(x_1) + f(x_0)}{2} \right ]\)

This is called trapezoidal rule and it is the area of the trapezoid whose width is (x1 – x0) and height is the average of f(x0) and f(x1)

Simpson’s 1/3 is an extension of trapezoidal rule where the integrand is approximated by a second order polynomial. The Simpson’s 1/3 rule assumes n = 2.

\(I = \int_{x_0}^{x_2}f(x) dx = \frac{h}{3} [f(x_0) + 4f(x_1) + f(x_2)]\)

This is called Simpson’s 1/3 rule.

Derivation of Simpson’s 1/3:

General quadrature formula for integration is given by

\(\int_{a}^{b}f(x)dx = \int_{x_0}^{x_0+nh}f(x)dx = nh\left [ f(x_0) + \frac{n}{2} Δf(x_0) + \frac{1}{12}(2n^2-3n)Δ^2f(x_0) + \frac{1}{24}(n^3-4n^2+4n)Δ^3f(x_0) + …… \right ]\)

Here, \(h = \frac{b-a}{n}\)

By putting n = 2 in the above relation and neglecting higher order forward difference written as

\(\int_{x_0}^{x_0 + nh}f(x)dx = \int_{x_0}^{x_2}f(x)dx = 2h\left [ f(x_0) + Δf(x_0) + \frac{1}{6}Δ^2f(x_0) \right ]\)
\(= 2h\left [ f(x_0) + (f(x_1) – f(x_0)) + \frac{1}{6}(f(x_0) – 2f(x_1) + f(x_2)) \right ]\)
\(= h\left [ 2f(x_0) + 2(f(x_1) – f(x_0)) + \frac{1}{3}(f(x_0) – 2f(x_1) + f(x_2)) \right ]\)
\(I = \int_{x_0}^{x_2}f(x) dx = \frac{h}{3} [f(x_0) + 4f(x_1) + f(x_2)]\)

This is called Simpson’s 1/3 rule.

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