Trapezoidal, Simpson’s, and Newton-Cotes rules are the three types of rules in the mathematical calculus to define and measure definite integrals and integrands as well. They are interrelated and helpful as effective methods for different valuation of integrals/integrands. The paper aims at discussing the gist of the rules and their historical background. Thus, it incorporates purely descriptive element in the body of the text so that any observer could make it out independently of his/her academic background.

## Trapezoidal Rule

To begin with, this rule touches closely upon the approximation of the definite integrals if one uses a particular application of calculus. This is a method of numerical calculation with an idea that an antiderivative is not overt; and it can be explained through the following formula, namely:

Lengths bases (a, b) and lengths height (b-a) are important when approximating integrals by means of the Trapezoidal rule (Buck, Wrkich, & Lamontagne, 2010). Needless to say, these valuations are to be first identified in terms of this method. In this case a mathematician uses this rule to measure the right trapezoid area and its relation to the height. Moreover, the trapezoidal rule overestimates the integral with the concave up function and, on the other side, it underestimates the integral with the concave down function (Trapezoidal Rule, 2011). In the historical cut, the method is credited to the works and times of Thomas Simpson (17th century) who then invented his own Simpson’s rule (O'Connor & Robertson, 2005).

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## Simpson’s Rule

In general, Simpson’s rule is a method to approximate definite integrals in the field of interpolation. It is closely referred to the Newton-Cotes formula where the approximation of the integral is done through using quadratic polynomials (parabolic arches):

This formula is appropriate in case when there is a particular “smoothness” of the function which means less oscillation possible for the intervals set between three prescribed points on the parable (Simpson's rule, 2011). Moreover, by using Simpson’s rule there is a possibility to get exact results when integrals of polynomials are integrated up to cubic degree (Weisstein, 2005). Thomas Simpson never claimed that this rule belongs to his invention. He just gave credit to Newton in his works (O'Connor & Robertson, 2005). To say more, this method was re-defined and applied to the modern mathematical calculus by Simpson, but, on the other hand, the idea was taken from Newton and Kepler (O'Connor & Robertson, 2005).

## Newton-Cotes Rule

This technique of numerical integration cannot be underestimated as it gives a full and clear picture on the integrals and integrands. Invented by Isaac Newton and Roger Cotes, it applies to evaluate the integrand at equally-spaced points realized through 1) “closed” (function value used at all points) and 2) “open” (function value not used at all points) types of formulae under this rule:

To make it plain, this rule is a set of formulae. Moreover, it has a straight-forward relation to the above mentioned two rules. However, the difference is that Trapezoid and Simpson’s rules are used for small values of n (Sanchez, 2010). Therefore, this rule is universal in the field of quadrature. It is a set of different methods to estimate the approximate value under the curve area (Weisstein, Newton-Cotes Formulas, 2005). Given that, Newton-Cotes rule includes the Trapezoidal and Simpson’s rules as it unites them with a little difference: the first rule is the 2-point closed Newton-Cotes formula; the second one is the 3-point Newton-Cotes formula (Weisstein, Newton-Cotes Formulas, 2005, p. 1).

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