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Simpson's 1/3 rule, also simply called Simpson's rule, is a method for numerical integration proposed by Thomas Simpson. It is based upon a quadratic interpolation and is the composite Simpson's 1/3 rule evaluated for n = 2 {\displaystyle n=2} .
Simpson's rule is used to find the approximate value of a definite integral by dividing the interval of integration into an even number of subintervals. Learn Simpson's 1/3 rule formula and its derivation with some examples.
Simpson's Rule is a numerical method that approximates the value of a definite integral by using quadratic functions. This method is named after the English mathematician Thomas Simpson (1710−1761).
Simpson’s rule estimates the area under the graph of f (x) by ap-proximating the function f (x) by a parabola and calculating the area. The figure above shows how Simpson’s rule approximates the area under the graph of a function f (x) by the area under. . under the parabola. Each parabolic approximation is.
Example: Use the Simpson’s rule with \(n = 10\) and compute ${\displaystyle \int_{0}^{1} x^2, dx.} Solution : Discretizing \([0,~1]\) using \(h = \dfrac{b-a}{n} = \dfrac{1 - 0}{10} = 0.1\) , \[\begin{equation*} \left\{ 0,~0.1,~0.2,~0.3,~0.4,~0.5,~0.6,~0.7,~0.8,~0.9,~1.0 \right\} \end{equation*}\]
Simpson’s Rule is based on the fact that given any three points, you can find the equation of a quadratic through those points. For example, let’s say you had points (3, 12), (1, 5), and (5, 9). Starting with (3, 12) and using y = ax2 + bx + c, you could write: x y. 12 = a(3)2 + b(3) + c.
Use Simpson’s rule to approximate the value of a definite integral to a given accuracy. With the midpoint rule, we estimated areas of regions under curves by using rectangles. In a sense, we approximated the curve with piecewise constant functions. With the trapezoidal rule, we approximated the curve by using piecewise linear functions.
