The area under the Stress S(e) curve from the initial position to the stretched position represents the work done by the

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The area under the Stress S(e) curve from the initial position
to the stretched position represents the work done by the spring
given by the function below. Based on the Newton’s interpolation in
problem (4), estimate the amount of work done from e1 = 0.5cm to e2
= 3.5cm using multiple application trapezoidal rule, Simpson’s 1/3
rule, Simpson’s 3/8 rule and Boole’s rule for n = 12 segments. Then
improve the trapezoidal Rule integral estimates using Romberg
integration for n = 6 and n = 12 segments. (Answers should be in 5
decimal places)