Textbook Question
81. Find the values of p for which each integral converges.
a. ∫ from 1 to 2 of [dx / (x (ln x)^p)]
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.7.17a
In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from 1 to 2 of 1/s² ds
Verified step by step guidance1
Identify the function to be integrated: \(f(x) = \frac{1}{x^2}\), and the interval of integration: \([1, 2]\).
Recall the error bound formula for the Trapezoidal Rule: \(|E_T| \leq \frac{(b - a)^3}{12 n^2} \max_{a \leq x \leq b} |f''(x)|\), where \(n\) is the number of subintervals.
Compute the second derivative of the function \(f(x)\). Start by finding the first derivative: \(f'(x) = -2 x^{-3}\), then the second derivative: \(f''(x) = 6 x^{-4}\).
Determine the maximum value of \(|f''(x)|\) on the interval \([1, 2]\). Since \(f''(x) = 6 / x^4\) is positive and decreasing on \([1, 2]\), the maximum occurs at \(x = 1\), so \(\max |f''(x)| = 6\).
Set up the inequality for the error bound to be less than \(10^{-4}\): \(\frac{(2 - 1)^3}{12 n^2} \times 6 < 10^{-4}\). Simplify and solve for \(n\) to find the minimum number of subintervals needed.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
8mWas this helpful?
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 accuracy depends on the number of subintervals; more subintervals generally yield better approximations.
Recommended video:
Guided course
5:50
Power Rules
Error Bound for the Trapezoidal Rule
The error bound for the Trapezoidal Rule estimates the maximum possible error in the approximation. It depends on the second derivative of the function, the length of the interval, and the number of subintervals. Specifically, the error is bounded by (b−a)³/(12n²) times the maximum of |f''(x)| on [a,b].
Recommended video:
04:57Determining Error and Relative Error
Second Derivative and Its Role in Error Estimation
The second derivative of the integrand measures the concavity of the function and influences the error in the Trapezoidal Rule. To estimate the error bound, one must find the maximum absolute value of the second derivative on the interval, as it directly affects how closely the trapezoids approximate the curve.
Recommended video:
06:02The Second Derivative Test: Finding Local Extrema
Related Practice
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.
a. u = 1/(x + 1)
What is the value of the integral?
Textbook Question
In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from -1 to 1 of (t³ + 1) dt
Textbook Question
Finding area
Find the area of the region enclosed by the curve y = x cos(x) and the x-axis (see the accompanying figure) for:
a. π/2 ≤ x ≤ 3π/2.
1
views
Textbook Question
In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from -1 to 1 of (x² + 1) dx
Textbook Question
Evaluate ∫ sec θ dθ by:
a. Multiplying by (sec θ + tan θ) / (sec θ + tan θ) and then using a u-substitution.