Answer to Question #283852 in Quantitative Methods for Ar-Ar

Question

You are given the following integral to evaluate: ∫(𝟑𝒙 − 𝟐)𝒅𝒙.

Without performing any calculations, which numerical method of the following gives the

best accuracy: (1) Trapezoidal rule, (2) Simpson’s 1/3 rule, and (3) Simpson’s 3/8 rule?

Justify your answer

Answer 1

When using trapezoidal rule,we take every interval as it is . When applying Simpson's rule, we further divide it into two parts and then apply the formula. Therefore, Simpson's is more precise.

Now when we come to Simpson's rule,we have the Simpson's 1/3 rule and Simpson's 3/8 rule.Simpson’s 1/3 rule is an extension of the trapezoidal rule in which the integrand is approximated by a second-order polynomial while the Simpson's 3/8 rule is entirely based on the cubic interpolation rather than the quadratic interpolation. The Simpson's 3/8 rule will give the best accuracy since it uses more functional values and as a result, the error term is smaller as compared to the Simpson's 1/3 rule.