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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 8 - Techniques of IntegrationProblema 8.7.2g
Capitolo 8, Problema 8.7.2g

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 3 of (2x - 1) dx

Guida verificata passo dopo passo
1
Identify the integral to approximate: \(\int_{1}^{3} (2x - 1) \, dx\) and note that \(n = 4\) subintervals will be used for Simpson's Rule.
Calculate the width of each subinterval using the formula \(\Delta x = \frac{b - a}{n} = \frac{3 - 1}{4} = 0.5\).
Determine the \(x\)-values at the endpoints and midpoints of the subintervals: \(x_0 = 1\), \(x_1 = 1.5\), \(x_2 = 2\), \(x_3 = 2.5\), and \(x_4 = 3\).
Evaluate the function \(f(x) = 2x - 1\) at each of these points: \(f(x_0)\), \(f(x_1)\), \(f(x_2)\), \(f(x_3)\), and \(f(x_4)\).
Apply Simpson's Rule formula: \(S_n = \frac{\Delta x}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4) \right]\) to estimate the integral, and then find the upper bound for the error \(|E_S|\) using the formula \(|E_S| \leq \frac{(b - a)^5}{180 n^4} \max |f^{(4)}(x)|\) on \([1,3]\).

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Simpson's Rule

Simpson's Rule is a numerical method for approximating definite integrals by dividing the interval into an even number of subintervals and fitting parabolas through the function values. It generally provides more accurate results than the Midpoint or Trapezoidal Rules for smooth functions.
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Error Bound for Simpson's Rule

The error bound for Simpson's Rule estimates the maximum possible difference between the exact 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.
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Determining Error and Relative Error

Definite Integral and Function Behavior

Understanding the definite integral as the area under the curve between two points is essential. Knowing the behavior of the function (here, linear: 2x - 1) helps in anticipating the accuracy of numerical methods and simplifies error estimation since higher derivatives may be zero.
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Definition of the Definite Integral
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Domanda del libro di testo

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 -2 to 0 of (x² - 1) dx

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Domanda del libro di testo

Using different substitutions

Show that the integral

∫((x² - 1)(x + 1))^(-2/3) dx

can be evaluated with any of the following substitutions.

f. u = arccos x

What is the value of the integral?

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Domanda del libro di testo

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 0 to π of sin(t) dt

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Domanda del libro di testo

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

13
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Domanda del libro di testo

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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Domanda del libro di testo

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

b. Evaluate the integral directly and find |ET|.

∫ from 1 to 3 of (2x - 1) dx

23
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