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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 8.7.1e
Chapter 8, Problem 8.7.1e

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

Verified step by step guidance
1
Identify the integral to evaluate: \(\int_{1}^{2} x \, dx\).
Evaluate the integral directly by finding the antiderivative of the function \(f(x) = x\). The antiderivative is \(F(x) = \frac{1}{2}x^{2}\).
Apply the Fundamental Theorem of Calculus by computing \(F(2) - F(1)\), which gives the exact value of the integral.
Recall the formula for the Trapezoidal Rule error bound \(|E_{T}| \leq \frac{(b - a)^{3}}{12n^{2}} \max_{a \leq x \leq b} |f''(x)|\), where \(n\) is the number of subintervals, \(a=1\), and \(b=2\).
Calculate the second derivative \(f''(x)\) of the function \(f(x) = x\), which is zero, and use this to find the exact error bound \(|E_{T}|\) for the Trapezoidal Rule.

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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 area under a curve into trapezoids rather than rectangles. It estimates the integral by averaging the function values at the endpoints of subintervals and multiplying by the subinterval width, providing a better approximation than simple Riemann sums.
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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 of x is (1/2)x², and evaluating it at the bounds gives the exact area under the curve.
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Definition of the Definite Integral

Error Estimation in Numerical Integration (|ET|)

Error estimation quantifies the difference between the exact integral value and its numerical approximation. For the Trapezoidal Rule, the error bound |ET| depends on the second derivative of the function and the width of the subintervals, helping assess the accuracy of the approximation.
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Estimating the Area Under a Curve with Right Endpoints & Midpoint
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

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.

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

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 1 / s² ds

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 3 of (2x - 1) dx