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 x dx

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Identify the integral to approximate: \(\int_{1}^{2} x \, dx\).

Determine the number of subintervals \(n = 4\), and calculate the width of each subinterval using \(\Delta x = \frac{b - a}{n} = \frac{2 - 1}{4} = 0.25\).

List the partition points: \(x_0 = 1\), \(x_1 = 1.25\), \(x_2 = 1.5\), \(x_3 = 1.75\), and \(x_4 = 2\).

Apply Simpson's Rule formula: \(S_n = \frac{\Delta x}{3} \left[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4)\right]\), where \(f(x) = x\) in this problem.

To find the error bound \(|E_S|\), use the formula: \(|E_S| \leq \frac{(b - a)^5}{180 n^4} \max_{a \leq x \leq b} |f^{(4)}(x)|\). Since \(f(x) = x\) is a polynomial of degree 1, its fourth derivative \(f^{(4)}(x) = 0\), so the error bound will be zero.

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주요 개념

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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 to provide a more accurate estimate than the Midpoint or Trapezoidal Rules.

Error Bound for Simpson's Rule

The error bound for Simpson's Rule estimates the maximum possible difference between the true integral and the approximation. It depends on the fourth 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 a Function

A definite integral calculates the net area under a curve between two points on the x-axis. Understanding the integral of the function f(x) = x from 1 to 2 involves knowing the antiderivative and the fundamental theorem of calculus, which connects integration and differentiation.

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 Rulea. Estimate the integral with n = 4 steps and find an upper bound for |ES|.∫ from 1 to 2 of x dx