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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 8.7.2e
Chapter 8, Problem 8.7.2e

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

Verified step by step guidance
1
First, identify the function to integrate: \(f(x) = 2x - 1\), and the interval of integration: from \(a = 1\) to \(b = 3\).
To evaluate the integral directly, set up the definite integral: \(\int_{1}^{3} (2x - 1) \, dx\).
Find the antiderivative of the function \(f(x)\). Since \(f(x) = 2x - 1\), its antiderivative is \(F(x) = x^{2} - x\).
Evaluate the antiderivative at the bounds and subtract: calculate \(F(3) - F(1)\), which gives the exact value of the integral.
To find the trapezoidal rule error bound \(|E_{T}|\), use the formula \(|E_{T}| \leq \frac{(b - a)^{3}}{12n^{2}} \max_{a \leq x \leq b} |f''(x)|\), where \(n\) is the number of subintervals and \(f''(x)\) is the second derivative of \(f(x)\). Compute \(f''(x)\) and substitute the values to find \(|E_{T}|\).

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Key Concepts

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

Definite Integral

A definite integral calculates the exact area under a curve between two limits. It is represented as ∫ from a to b of f(x) dx, where a and b are the interval bounds. Evaluating it directly involves finding the antiderivative and computing its difference at the bounds.
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Definition of the Definite Integral

Trapezoidal Rule

The Trapezoidal Rule is a numerical method to approximate definite integrals by dividing the area under the curve into trapezoids. It estimates the integral by summing the areas of these trapezoids, which are formed by connecting points on the function with straight lines.
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Power Rules

Trapezoidal Rule Error Bound (|ET|)

The error bound |ET| measures the difference between the exact integral and the Trapezoidal Rule approximation. It depends on the second derivative of the function and the width of the subintervals, providing a way to estimate the accuracy of the numerical approximation.
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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.

II. Using the Trapezoidal Rule

b. Evaluate the integral directly and find |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

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

∫ from 0 to π of sin(t) dt

Textbook Question

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?

1
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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

Using different substitutions

Show that the integral

∫((x² - 1)(x + 1))^(-2/3) dx

can be evaluated with any of the following substitutions.

e. u = tan^(-1) ((x - 1)/2)

What is the value of the integral?

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 x dx