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Question 4.7.8: Comparing the Midpoint, Trapezoidal and Simpson’s Rules Comp...

Comparing the Midpoint, Trapezoidal and Simpson’s Rules

Compute the Midpoint, Trapezoidal and Simpson’s Rule approximations of \int_0^1 \frac{4}{x^2+1} d x with n = 10, n = 20, n = 50 and n = 100. Compare to the exact value of π.

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n Midpoint Rule Trapezoidal Rule Simpson’s Rule
10 3.142425985 3.139925989 3.141592614
20 3.141800987 3.141175987 3.141592653
50 3.141625987 3.141525987 3.141592654
100 3.141600987 3.141575987 3.141592654

 

Compare these values to the exact value of π ≈ 3.141592654. Note that the Midpoint Rule tends to be slightly closer to π than the Trapezoidal Rule, but neither is as close with n = 100 as Simpson’s Rule is with n = 10.

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