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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 8.7.7e
Chapter 8, Problem 8.7.7e

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

Verified step by step guidance
1
First, identify the integral to evaluate: \(\int_{1}^{2} \frac{1}{s^{2}} \, ds\).
To evaluate the integral directly, rewrite the integrand as \(s^{-2}\) and find its antiderivative. Recall that the antiderivative of \(s^{n}\) is \(\frac{s^{n+1}}{n+1}\) for \(n \neq -1\).
Compute the antiderivative: \(\int s^{-2} \, ds = \frac{s^{-1}}{-1} = -s^{-1} = -\frac{1}{s}\).
Evaluate the definite integral by applying the Fundamental Theorem of Calculus: calculate \(-\frac{1}{s}\) at the upper limit \(s=2\) and subtract its value at the lower limit \(s=1\).
To find the error bound \(|E_{T}|\) for the Trapezoidal Rule, use the formula \(|E_{T}| \leq \frac{(b - a)^{3}}{12 n^{2}} \max_{a \leq s \leq b} |f''(s)|\), where \(f(s) = \frac{1}{s^{2}}\). Compute the second derivative \(f''(s)\), find its maximum absolute value on \([1,2]\), and substitute all values into the formula.

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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 the integral directly involves finding the antiderivative and applying the Fundamental Theorem of Calculus.
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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 is useful when the integral is difficult to evaluate analytically.
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Trapezoidal Rule Error Bound (|ET|)

The error bound |ET| for the Trapezoidal Rule estimates the maximum difference between the exact integral and its trapezoidal approximation. It depends on the second derivative of the function and the width of the subintervals, providing insight into the accuracy of the 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

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

∫ 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.

II. Using the Trapezoidal Rule

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

∫ 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.

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 1 to 2 of 1 / s² ds