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.
II. Using the Trapezoidal Rule
a. Estimate the integral with n = 4 steps and find an upper bound for |ET|.
∫ from 1 to 2 of 1 / s² ds

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

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 the Trapezoidal Rule formula: \(T_n = \frac{h}{2} \left[f(s_0) + 2f(s_1) + 2f(s_2) + 2f(s_3) + f(s_4)\right]\), where you substitute the function values at each partition point.

To find the upper bound for the error \(|E_T|\), use the error bound formula for the Trapezoidal Rule: \(|E_T| \leq \frac{(b - a)^3}{12 n^2} \max_{a \leq s \leq b} |f''(s)|\). Calculate the second derivative \(f''(s)\), find its maximum absolute value on \([1, 2]\), and substitute all values to estimate the error bound.

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주요 개념

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Trapezoidal Rule

The Trapezoidal Rule is a numerical method to approximate definite integrals by dividing the interval into subintervals and approximating the area under the curve as trapezoids. The sum of these trapezoidal areas provides an estimate of the integral, improving accuracy with more subintervals (n).

Error Bound for the Trapezoidal Rule

The error bound |ET| for the Trapezoidal Rule estimates the maximum possible difference between the true integral and its approximation. It depends on the second derivative of the function, the interval length, and the number of subintervals, providing a way to assess the accuracy of the approximation.

Definite Integral of 1/s² from 1 to 2

The integral ∫₁² 1/s² ds represents the area under the curve y = 1/s² between s = 1 and s = 2. Understanding the behavior and derivatives of this function is essential for applying numerical methods and calculating error bounds accurately.

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.II. Using the Trapezoidal Rulea. Estimate the integral with n = 4 steps and find an upper bound for |ET|.∫ from 1 to 2 of 1 / s² ds