The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.
III. Using Simpson's Rule
a. Estimate the integral with n = 4 steps and find an upper bound for |ES|.
∫ from 1 to 2 of 1 / s² ds

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Identify the function to integrate: \(f(s) = \frac{1}{s^2} = s^{-2}\), over the interval \([1, 2]\).

Calculate the step size \(h\) using the formula \(h = \frac{b - a}{n}\), where \(a = 1\), \(b = 2\), and \(n = 4\).

Determine the partition points: \(s_0 = 1\), \(s_1 = 1 + h\), \(s_2 = 1 + 2h\), \(s_3 = 1 + 3h\), and \(s_4 = 2\).

Apply Simpson's Rule formula for \(n=4\) steps: \[ \int_1^2 f(s) \, ds \approx \frac{h}{3} \left[f(s_0) + 4f(s_1) + 2f(s_2) + 4f(s_3) + f(s_4)\right] \] Calculate each \(f(s_i)\) and substitute into the formula.

To find the upper bound for the error \(|E_S|\), use the error bound formula for Simpson's Rule: \[ |E_S| \leq \frac{(b - a)^5}{180 n^4} \max_{a \leq s \leq b} |f^{(4)}(s)| \] Find the fourth derivative \(f^{(4)}(s)\) of \(f(s)\), determine its maximum absolute value on \([1, 2]\), and substitute all values into the error bound formula.

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Simpson's Rule

Simpson's Rule is a numerical method for approximating definite integrals by fitting parabolas through segments of the function. It requires an even number of subintervals (n) and combines the function values at endpoints and midpoints with specific weights to estimate the integral more accurately than the Midpoint or Trapezoidal Rules.

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Error Bound for Simpson's Rule

The error bound for Simpson's Rule estimates the maximum possible difference between the true integral and its approximation. It depends on the fourth derivative of the function over the interval, the length of the interval, and the number of subintervals n. This bound helps assess the accuracy of the approximation.

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The integral ∫₁² 1/s² ds represents the area under the curve y = 1/s² between s = 1 and s = 2. Understanding the behavior of this function, which is positive and decreasing, is important for applying numerical methods and estimating errors effectively.

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