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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 8.7.2d
Chapter 8, Problem 8.7.2d

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 3 of (2x - 1) dx

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1
Identify the integral to approximate: \(\int_{1}^{3} (2x - 1) \, dx\) and note that we will use the Trapezoidal Rule with \(n = 4\) subintervals.
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 of each subinterval: \(x_0 = 1\), \(x_1 = 1.5\), \(x_2 = 2\), \(x_3 = 2.5\), and \(x_4 = 3\).
Evaluate the function \(f(x) = 2x - 1\) at each of these points to get \(f(x_0), f(x_1), f(x_2), f(x_3), f(x_4)\).
Apply the Trapezoidal Rule formula: \(T_n = \frac{\Delta x}{2} \left[f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4)\right]\) to estimate the integral.

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Key Concepts

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

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 trapezoid areas provides an estimate of the integral, improving accuracy with more subintervals (n).
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Power Rules

Error Bound for the Trapezoidal Rule

The error bound |ET| estimates the maximum possible difference between the true integral value and the Trapezoidal Rule 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.
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Definite Integral of a Linear Function

Integrating a linear function like 2x - 1 over an interval can be done exactly using antiderivatives. Understanding this helps verify numerical approximations and error bounds, as the integral of a linear function is straightforward and the second derivative is zero, affecting error estimates.
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Definition of the Definite Integral
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.

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 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.

II. Using the Trapezoidal Rule

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

∫ 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.

II. Using the Trapezoidal Rule

a. Estimate the integral with n = 4 steps and find an upper bound for |ET|.

∫ from 0 to 2 of (t³ + 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.

II. Using the Trapezoidal Rule

a. Estimate the integral with n = 4 steps and find an upper bound for |ET|.

∫ 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.

II. Using the Trapezoidal Rule

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

∫ 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.

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