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Numerical Integration Using the Trapezoidal Rule and Simpson’s Rule

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Calculate the integral over the finite interval (a, b) using the trapezoidal rule, the midpoint rule, and Simpson’s rule.
The number of subdivisions is doubled at each step: 2, 4, 8, 16, ..., N. The integrand f(x) is assumed to be analytic and nonperiodic.
f(x)
a
,
b
Maximum number of divisions: N
\(\normalsize \hspace{40px}S={\large\int_a^{b}}f(x)dx\\ (1)\ Trapezoid\hspace{5px}S_{\normalsize t}={\large\frac{b-a}{2}}\{f(a)+f(b)\}\\ (2)\ Midpoint\hspace{10px}S_m=(b-a)f({\large\frac{a+b}{2}})\\ (3)\ Simpson\hspace{10px}S_{\normalsize s}={\large\frac{S_{\normalsize t}+2S_m}{3}}\\ \hspace{105px}={\large\frac{b-a}{6}}\Biggl[f(a)+4f({\large\frac{a+b}{2}})+f(b)\Biggr]\\ \)

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Numerical Integration Using the Trapezoidal Rule and Simpson’s Rule | Keisan