MathsSimpson’s Rule – Explanation, Graphical Representation and FAQs

Simpson’s Rule – Explanation, Graphical Representation and FAQs

Simpson’s Rule Formula

Simpson’s Rule is a mathematical formula used to calculate the numerical value of a definite integral. The formula is used to approximate the value of an integral by breaking the integral down into a series of smaller integrals, each of which is approximated using a different Simpson’s Rule formula. The final value of the integral is then the sum of the individual Simpson’s Rule values.

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    Mathematically, Simpson’s Rule is a formula used to approximate the definite integral of a function. The integral is approximated using a weighted average of function values at points within the given interval. Simpson’s Rule is more accurate than the Trapezoidal Rule, which is another formula used to approximate definite integrals. Simpson’s Rule is named after Edward Simpson, who developed the formula in 1817.

    Graphical Representation

    The Simpson’s Rule formula is:

    where “n” is the number of sub-intervals, “x” is the point within the interval at which the value is to be calculated, and “f” is the function being integrated.

    A graphical representation of Simpson’s Rule can be helpful in understanding how it works. In the diagram below, the area under the curve is divided into four equal sub-intervals, and the value of “f” at “x” is shown for each sub-interval. The average of these values is then calculated, and this is the value that is used to calculate the area under the curve.

    In the example shown, the area under the curve is 3.14. The Simpson’s Rule formula would calculate this as:

    where “n” is 4 (the number of sub-intervals), “x” is the point within the interval at which the value is to be calculated, and “f” is the function being integrated.

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