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Why should I use the simpson or trapezium rule when calculating the area under a curve? It is much easier and accurate when using integration the ordinary way
The Simpson's rule is a numerical method used for approximating the area under a curve. It uses a combination of quadratic polynomials to create a more accurate approximation compared to the trapezium rule.
The Simpson's rule works by dividing the area under the curve into a series of parabolas, creating an even more accurate approximation compared to the trapezium rule. It uses an odd number of equally spaced intervals to create the parabolas.
The Simpson's rule is typically used when the function being integrated is a smooth curve, rather than a series of straight lines. It is also most effective when the number of intervals is large.
The main difference between the Simpson's rule and the trapezium rule is the number of intervals used in the approximation. The Simpson's rule uses an odd number of intervals and creates parabolas, while the trapezium rule uses an even number of intervals and creates trapezoids.
Like any numerical method, the Simpson's rule has its limitations. It may not provide an accurate approximation for functions that are not smooth curves. It also requires a large number of intervals to be effective, which can be time-consuming to calculate.