The Simpson's Rule formula is given by:
A ≈ (h/3) * [f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + 2f(x4) + ... + 2f(xn-2) + 4f(xn-1) + f(xn)]
where h is the width of each partition, n is the number of partitions, and f(x) is the function being integrated.
The more partitions used in Simpson's Rule, the more accurate the approximation of the area under the curve will be. As the number of partitions approaches infinity, the approximation becomes more and more precise.
Yes, Simpson's Rule can be used for any continuous function. However, it may not always be the most accurate method for finding the area under the curve. Other methods such as the Trapezoidal Rule may be more appropriate for certain functions.
The value of h is typically chosen by dividing the interval of integration into equal subintervals. The smaller the value of h, the more accurate the approximation will be. However, using too small of a value for h can also introduce rounding errors.
Simpson's Rule is generally more accurate than other methods such as the Trapezoidal Rule, especially for functions that have a more complex shape. It also requires fewer evaluations of the function, making it more efficient for calculating the area under the curve.