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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.7.4g
Chapter 8, Problem 8.7.4g

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

Verified step by step guidance
1
Identify the integral to approximate: \(\int_{-2}^{0} (x^{2} - 1) \, dx\) and note that \(n = 4\) subintervals will be used for Simpson's Rule.
Calculate the width of each subinterval using the formula \(\Delta x = \frac{b - a}{n}\), where \(a = -2\) and \(b = 0\).
Determine the \(x\)-values (nodes) at which the function will be evaluated: \(x_0 = -2\), \(x_1 = -2 + \Delta x\), \(x_2 = -2 + 2\Delta x\), \(x_3 = -2 + 3\Delta x\), and \(x_4 = 0\).
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 \(f(x) = x^{2} - 1\).
To find the upper bound for the error \(|E_S|\), use the error bound formula for Simpson's Rule: \(|E_S| \leq \frac{(b - a)^5}{180 n^4} \max_{a \leq x \leq b} |f^{(4)}(x)|\) Calculate the fourth derivative \(f^{(4)}(x)\) of the function and find its maximum absolute value on the interval \([-2, 0]\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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. The formula combines function values at equally spaced points with specific weights to estimate the integral.
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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 the approximation. It depends on the fourth derivative of the function over the interval and the number of subintervals n. Specifically, the error bound is proportional to the maximum absolute value of the fourth derivative and inversely proportional to n to the fourth power, ensuring accuracy improves with more steps.
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Definite Integral and Function Behavior

Understanding the definite integral involves interpreting the area under the curve of the function between given limits. For the function f(x) = x² - 1 on [-2, 0], knowing its shape and derivatives helps in applying numerical methods and error analysis. Recognizing the function's smoothness and polynomial nature simplifies calculating derivatives needed for error bounds.
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Related Practice
Textbook Question

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

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

∫ from 0 to π of sin(t) dt

Textbook Question

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

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

∫ from 1 to 2 of 1 / s² ds

Textbook Question

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

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

∫ from 1 to 3 of (2x - 1) dx

Textbook Question

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

Textbook Question

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

Textbook Question

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 2 of (t³ + t) dt