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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 8.7.1i
Chapter 8, Problem 8.7.1i

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 x dx

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1
Identify the integral to be evaluated: \(\int_{1}^{2} x \, dx\).
Recall the formula for the exact value of a definite integral: \(\int_{a}^{b} f(x) \, dx = F(b) - F(a)\), where \(F(x)\) is an antiderivative of \(f(x)\).
Find the antiderivative of \(f(x) = x\). Since the derivative of \(\frac{1}{2}x^{2}\) is \(x\), we have \(F(x) = \frac{1}{2}x^{2}\).
Evaluate the definite integral by substituting the limits: calculate \(F(2) - F(1) = \frac{1}{2} \times 2^{2} - \frac{1}{2} \times 1^{2}\).
To find the error bound \(|E_{S}|\) for Simpson's Rule, recall the error 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 is zero, so the error bound \(|E_{S}|\) is zero.

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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.
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Power Rules

Exact Evaluation of Definite Integrals

Exact evaluation involves finding the precise value of a definite integral using antiderivatives and the Fundamental Theorem of Calculus. For ∫ from 1 to 2 of x dx, the antiderivative is (1/2)x², evaluated at the limits to get the exact area.
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Definition of the Definite Integral

Error Bound in Simpson's Rule (|ES|)

The error bound |ES| estimates the maximum difference between the exact integral and the Simpson's Rule approximation. It depends on the fourth derivative of the function and the width of the subintervals, providing insight into the accuracy of the numerical approximation.
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Intro to the Chain Rule Example 1
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 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 -2 to 0 of (x² - 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

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

∫ from 1 to 3 of (2x - 1) dx